FORMALIZED MATHEMATICS Volume 12, Number 2, 2004 University of Białystok

Banach Space of Bounded Real Sequences

Yasumasa Suzuki Take, Yokosuka-shi Japan

Summary. We introduce the arithmetic addition and multiplication in the set of bounded real sequences and also introduce the norm. This set has the structure of the Banach space.

MML Identifier: RSSPACE4.

The articles [23], [6], [27], [29], [28], [15], [21], [3], [1], [2], [20], [24], [9], [4], [5], [7], [26], [22], [16], [17], [14], [11], [12], [10], [25], [13], [8], [19], and [18] provide the notation and terminology for this paper.

1. The Banach Space of Bounded Real Sequences

The subset the set of bounded real sequences of the linear space of real sequences is defined by the condition (Def. 1).

(Def. 1) Let x be a set. Then x ∈ the set of bounded real sequences if and only

if x ∈ the set of real sequences and id_{seq}(x) is bounded. Let us note that the set of bounded real sequences is non empty and the set

of bounded real sequences is linearly closed.

One can prove the following proposition

(1) ⟨the set of bounded real sequences, Zero (the set of bounded real sequences, the linear space of real sequences), Add (the set of bounded real sequences, the linear space of real sequences), Mult (the set of bounded real sequences, the linear space of real sequences)⟩ is a subspace of the linear space of real sequences.

One can verify that ⟨the set of bounded real sequences, Zero (the set of bounded real sequences, the linear space of real sequences), Add (the set of bounded real sequences, the linear space of real sequences), Mult (the set of bounded real sequences, the linear space of real sequences)⟩ is Abelian, add-associative, right zeroed, right complementable, and real linear space-like.

The function linfty-norm from the set of bounded real sequences into ℝ is defined by:

(Def. 2) For every set x such that x ∈ the set of bounded real sequences holds

linfty-norm(x) = suprng|id_{seq}(x)|. The following proposition is true

(2) Let r_{1} be a sequence of real numbers. Then r_{1} is bounded and suprng|r_{1}| = 0 if and only if for every natural number n holds r_{1}(n) = 0.

Let us mention that ⟨the set of bounded real sequences, Zero (the set of bounded real sequences, the linear space of real sequences), Add (the set of bounded real sequences, the linear space of real sequences), Mult (the set of bounded real sequences, the linear space of real sequences), linfty-norm⟩ is Abelian, add-associative, right zeroed, right complementable, and real linear space-like.

The non empty normed structure linfty-Space is defined by the condition (Def. 3).

(Def. 3) linfty-Space = ⟨the set of bounded real sequences, Zero (the set of bounded real sequences, the linear space of real sequences), Add (the set of bounded real sequences, the linear space of real sequences), Mult (the set of bounded real sequences, the linear space of real sequences), linfty-norm⟩.

We now state two propositions:

(3) The carrier of linfty-Space = the set of bounded real sequences and for every set x holds x is a vector of linfty-Space iff x is a sequence of real numbers and id_{seq}(x) is bounded and 0_{linfty-Space} = Zeroseq and for every vector u of linfty-Space holds u = id_{seq}(u) and for all vectors u, v of linfty-Space holds u + v = id_{seq}(u) + id_{seq}(v) and for every real number r and for every vector u of linfty-Space holds r · u = r id_{seq}(u) and for every vector u of linfty-Space holds − u = − id_{seq}(u) and id_{seq}(− u) = − id_{seq}(u) and for all vectors u, v of linfty-Space holds u−v = id_{seq}(u)−id_{seq}(v) and for every vector v of linfty-Space holds id_{seq}(v) is bounded and for every vector v of linfty-Space holds ∥v∥ = suprng|id_{seq}(v)|.

(4) Let x, y be points of linfty-Space and a be a real number. Then ∥x∥ = 0 iff x = 0_{linfty-Space} and 0 ∥x∥ and ∥x+y∥ ∥x∥+∥y∥ and ∥a·x∥ = |a|·∥x∥.

Let us observe that linfty-Space is real normed space-like, real linear spacelike, Abelian, add-associative, right zeroed, and right complementable.

Next we state the proposition

(5) For every sequence v_{1} of linfty-Space such that v_{1} is Cauchy sequence by norm holds v_{1} is convergent.

2. The Banach Space of Bounded Functions

Let X be a non empty set, let Y be a real normed space, and let I_{1} be a function from X into the carrier of Y. We say that I_{1} is bounded if and only if:

(Def. 4) There exists a real number K such that 0 K and for every element x

of X holds ∥I_{1}(x)∥ K. The following proposition is true

(6) Let X be a non empty set, Y be a real normed space, and f be a function from X into the carrier of Y. If for every element x of X holds f(x) = 0_{Y} , then f is bounded.

Let X be a non empty set and let Y be a real normed space. Note that there exists a function from X into the carrier of Y which is bounded.

Let X be a non empty set and let Y be a real normed space. The functor BdFuncs(X, Y) yields a subset of RealVectSpace(X, Y) and is defined by:

(Def. 5) For every set x holds x ∈ BdFuncs(X, Y) iff x is a bounded function

from X into the carrier of Y. Let X be a non empty set and let Y be a real normed space. Observe that

BdFuncs(X, Y) is non empty.

The following propositions are true:

(7) For every non empty set X and for every real normed space Y holds BdFuncs(X, Y) is linearly closed.

(8) For every non empty set X and for every real normed space Y holds ⟨BdFuncs(X, Y), Zero (BdFuncs(X, Y), RealVectSpace(X, Y)), Add (BdFuncs(X, Y), RealVectSpace(X, Y)), Mult (BdFuncs(X, Y), RealVectSpace(X, Y))⟩ is a subspace of RealVectSpace(X, Y).

Let X be a non empty set and let Y be a real normed space. One can verify that ⟨BdFuncs(X, Y), Zero (BdFuncs(X, Y), RealVectSpace(X, Y)),

Add (BdFuncs(X, Y), RealVectSpace(X, Y)), Mult (BdFuncs(X, Y), RealVectSpace(X, Y))⟩ is Abelian, add-associative, right zeroed, right com-

plementable, and real linear space-like.

One can prove the following proposition

(9) For every non empty set X and for every real normed space Y holds ⟨BdFuncs(X, Y), Zero (BdFuncs(X, Y), RealVectSpace(X, Y)), Add (BdFuncs(X, Y), RealVectSpace(X, Y)), Mult (BdFuncs(X, Y), RealVectSpace(X, Y))⟩ is a real linear space.

Let X be a non empty set and let Y be a real normed space. The set of bounded real sequences from X into Y yields a real linear space and is defined as follows:

(Def. 6) The set of bounded real sequences from X into Y = ⟨BdFuncs(X, Y), Zero (BdFuncs(X, Y), RealVectSpace(X, Y)), Add (BdFuncs(X, Y),

RealVectSpace(X, Y)), Mult (BdFuncs(X, Y), RealVectSpace(X, Y))⟩. Let X be a non empty set and let Y be a real normed space. Observe that

the set of bounded real sequences from X into Y is strict.

One can prove the following three propositions:

(10) Let X be a non empty set, Y be a real normed space, f, g, h be vectors of the set of bounded real sequences from X into Y, and f′, g′, h′be bounded functions from X into the carrier of Y. Suppose f′ = f and g′ = g and h′ = h. Then h = f + g if and only if for every element x of X holds h′(x) = f′(x) + g′(x).

(11) Let X be a non empty set, Y be a real normed space, f, h be vectors of the set of bounded real sequences from X into Y, and f′, h′ be bounded functions from X into the carrier of Y. Suppose f′= f and h′= h. Let a be a real number. Then h = a · f if and only if for every element x of X holds h′(x) = a · f′(x).

(12) Let X be a non empty set and Y be a real normed space. Then

0_{the set of bounded real sequences from X into Y} = X ⟼→ 0_{Y} . Let X be a non empty set, let Y be a real normed space, and let f be a set.

Let us assume that f ∈ BdFuncs(X, Y). The functor modetrans(f, X, Y) yields a bounded function from X into the carrier of Y and is defined as follows:

(Def. 7) modetrans(f, X, Y) = f.

Let X be a non empty set, let Y be a real normed space, and let u be a function from X into the carrier of Y. The functor PreNorms(u) yielding a non empty subset of ℝ is defined as follows:

(Def. 8) PreNorms(u) = {∥u(t)∥ : t ranges over elements of X}.

Next we state three propositions:

(13) Let X be a non empty set, Y be a real normed space, and g be a bounded function from X into the carrier of Y. Then PreNorms(g) is non empty and upper bounded.

(14) Let X be a non empty set, Y be a real normed space, and g be a function from X into the carrier of Y. Then g is bounded if and only if PreNorms(g) is upper bounded.

(15) Let X be a non empty set and Y be a real normed space. Then there exists a function N_{1} from BdFuncs(X, Y) into ℝ such that for every set f if f ∈ BdFuncs(X, Y), then N_{1}(f) = supPreNorms(modetrans(f, X, Y)).

Let X be a non empty set and let Y be a real normed space. The functor BdFuncsNorm(X, Y) yielding a function from BdFuncs(X, Y) into ℝ is defined by:

(Def. 9) For every set x such that x ∈ BdFuncs(X, Y) holds

BdFuncsNorm(X, Y)(x) = supPreNorms(modetrans(x, X, Y)). One can prove the following two propositions:

(16) Let X be a non empty set, Y be a real normed space, and f be a bounded function from X into the carrier of Y. Then modetrans(f, X, Y) = f.

(17) Let X be a non empty set, Y be a real normed space, and f be a bounded function from X into the carrier of Y. Then BdFuncsNorm(X, Y)(f) = supPreNorms(f).

Let X be a non empty set and let Y be a real normed space. The real normed space of bounded functions from X into Y yielding a non empty normed structure is defined as follows:

(Def. 10) The real normed space of bounded functions from X into Y = ⟨BdFuncs(X, Y), Zero (BdFuncs(X, Y), RealVectSpace(X, Y)), Add (BdFuncs(X, Y), RealVectSpace(X, Y)), Mult (BdFuncs(X, Y), RealVectSpace(X, Y)), BdFuncsNorm(X, Y)⟩.

We now state several propositions:

(18) Let X be a non empty set and Y be a real normed space. Then X ⟼→ 0_{Y} = 0_{the real normed space of bounded functions from X into Y} .

(19) Let X be a non empty set, Y be a real normed space, f be a point of the real normed space of bounded functions from X into Y, and g be a bounded function from X into the carrier of Y. If g = f, then for every element t of X holds ∥g(t)∥ ∥f∥.

(20) Let X be a non empty set, Y be a real normed space, and f be a point of the real normed space of bounded functions from X into Y. Then 0 ∥f∥.

(21) Let X be a non empty set, Y be a real normed space, and f be a point of the real normed space of bounded functions from X into Y. Suppose f = 0_{the real normed space of bounded functions from X into Y} . Then 0 = ∥f∥.

(22) Let X be a non empty set, Y be a real normed space, f, g, h be points of the real normed space of bounded functions from X into Y, and f′, g′, h′be bounded functions from X into the carrier of Y. Suppose f′= f and g′= g and h′= h. Then h = f + g if and only if for every element x of X holds h′(x) = f′(x) + g′(x).

(23) Let X be a non empty set, Y be a real normed space, f, h be points of the real normed space of bounded functions from X into Y, and f′, h′ be bounded functions from X into the carrier of Y. Suppose f′ = f and h′ = h. Let a be a real number. Then h = a · f if and only if for every element x of X holds h′(x) = a · f′(x).

(24) Let X be a non empty set, Y be a real normed space, f, g be points of the real normed space of bounded functions from X into Y, and a be a real number. Then

(i) ∥f∥ = 0 iff f = 0_{the real normed space of bounded functions from X into Y} , (ii) ∥a · f∥ = |a| · ∥f∥, and (iii) ∥f + g∥ ∥f∥ + ∥g∥.

(25) Let X be a non empty set and Y be a real normed space. Then the real normed space of bounded functions from X into Y is real normed space-like.

(26) Let X be a non empty set and Y be a real normed space. Then the real

normed space of bounded functions from X into Y is a real normed space. Let X be a non empty set and let Y be a real normed space. Observe

that the real normed space of bounded functions from X into Y is real normed space-like, real linear space-like, Abelian, add-associative, right zeroed, and right complementable.

We now state three propositions:

(27) Let X be a non empty set, Y be a real normed space, f, g, h be points of the real normed space of bounded functions from X into Y, and f′, g′, h′be bounded functions from X into the carrier of Y. Suppose f′= f and g′= g and h′= h. Then h = f − g if and only if for every element x of X holds h′(x) = f′(x) − g′(x).

(28) Let X be a non empty set and Y be a real normed space. Suppose Y is complete. Let s_{1} be a sequence of the real normed space of bounded functions from X into Y. If s_{1} is Cauchy sequence by norm, then s_{1} is convergent.

(29) Let X be a non empty set and Y be a real Banach space. Then the real

normed space of bounded functions from X into Y is a real Banach space. Let X be a non empty set and let Y be a real Banach space. One can verify

that the real normed space of bounded functions from X into Y is complete.

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Solving Roots of Polynomial Equation of Degree 2 and 3 with Complex Coefficients

Yuzhong Ding QingDao University of Science and Technology

Xiquan Liang QingDao University of Science and Technology

Summary. In the article, solving complex roots of polynomial equation of degree 2 and 3 with real coefficients and complex roots of polynomial equation of degree 2 and 3 with complex coefficients is discussed.

MML Identifier: POLYEQ 3.

The terminology and notation used here are introduced in the following articles: [20], [15], [2], [5], [3], [8], [17], [16], [14], [10], [12], [7], [18], [1], [13], [21], [9], [19], [11], [6], and [4].

1. Solving Complex Roots of Polynomial Equation of Degree 2 and 3 with Real Coefficients

We follow the rules: a, b, c, d, a′, b′, c′, d′, x, y, x_{1}, u, v are real numbers and s, t, h, z, z_{1}, z_{2}, z_{3}, z_{4}, s_{1}, s_{2}, s_{3}, p, q are elements of ℂ.

Let a be a real number and let us consider z. Then a · z is an element of ℂ and it can be characterized by the condition:

(Def. 1) a · z = (a + 0i) · z.

Then a + z is an element of ℂ and it can be characterized by the condition: (Def. 2) a + z = z + (a + 0i).

Let us consider z. Then z^{2} is an element of ℂ and it can be characterized by the condition:

(Def. 3) z^{2} = (ℜ(z)^{2} − ℑ(z)^{2}) + (2 · (ℜ(z) · ℑ(z)))i. Let us consider a, b, c, z. Then Poly2(a, b, c, z) is an element of ℂ. The following propositions are true:

(1) (a + ci) · (b + di) = (a · b − c · d) + (a · d + b · c)i.

(2) If z = x + yi, then z^{2} = (x^{2} − y^{2}) + (2 · x · y)i.

(3) For all a, b holds (a + 0i) · (b + 0i) = a · b + 0i.

(4) If a ≠ 0 and Δ(a, b, c) 0 and Poly2(a, b, c, z) = 0, then z = ^{\frac{− b+\sqrt{Δ(a,b,c)}}{2·a}} or z = ^{\frac{− b−\sqrt{Δ(a,b,c)}}{2·a}} or z = − \frac{b}{2·a}.

(5) If a ≠ 0 and Δ(a, b, c) < 0 and Poly2(a, b, c, z) = 0_{ℂ}, then z = − \frac{b}{2·a} + \frac{\sqrt{− Δ(a,b,c)}}{2·a}i or z = − \frac{b}{2·a}+ (− ^{\frac{\sqrt{− Δ(a,b,c)}}{2·a}})i.

(6) If b ≠ 0 and for every z holds Poly2(0, b, c, z) = 0_{ℂ}, then z = − \frac{c}{b}.

(7) Let a, b, c be real numbers and z, z_{1}, z_{2} be elements of ℂ. Suppose a ≠ 0. Suppose that for every element z of ℂ holds Poly2(a, b, c, z) = Quard(a, z_{1}, z_{2}, z). Then \frac{b}{a} + 0i = − (z_{1} + z_{2}) and \frac{c}{a} + 0i = z_{1} · z_{2}.

Let z be an element of ℂ. The functor z^{3} yielding an element of ℂ is defined by:

(Def. 4) z^{3} = z^{2} · z.

Let a, b, c, d be real numbers and let z be an element of ℂ. The functor Poly_{3}(a, b, c, d, z) yielding an element of ℂ is defined as follows:

(Def. 5) Poly_{3}(a, b, c, d, z) = a · z^{3} + b · z^{2} + c · z + d. We now state a number of propositions:

(8) (0_{ℂ})^{3} = 0_{ℂ}.

(9) (1_{ℂ})^{3} = 1_{ℂ}.

(10) (− 1_{ℂ})^{3} = − 1_{ℂ}.

(11) ℜ(z^{3}) = ℜ(z)^{3} − 3 · ℜ(z) · ℑ(z)^{2} and ℑ(z^{3}) = − ℑ(z)^{3} + 3 · ℜ(z)^{2} · ℑ(z).

(12) If for every z holds Poly_{3}(a, b, c, d, z) = Poly_{3}(a′, b′, c′, d′, z), then a = a′ and b = b′ and c = c′ and d = d′.

(13) (z + h)^{3} = z^{3} + 3 · h · z^{2} + 3 · h^{2} · z + h^{3}.

(14) (z · h)^{3} = z^{3} · h^{3}.

(15) If b ≠ 0 and Poly_{3}(0, b, c, d, z) = 0_{ℂ} and Δ(b, c, d) 0, then z = \frac{− c+\sqrt{Δ(b,c,d)}}{2·b} or z = ^{\frac{− c−\sqrt{Δ(b,c,d)}}{2·b}} or z = − \frac{c}{2·b}.

(16) If b ≠ 0 and Poly_{3}(0, b, c, d, z) = 0_{ℂ} and Δ(b, c, d) < 0, then z = − \frac{c}{2·b}+ \frac{\sqrt{− Δ(b,c,d)}}{2·b}i or z = − \frac{c}{2·b}+ (− ^{\frac{\sqrt{− Δ(b,c,d)}}{2·b}})i.

(17) If a ≠ 0 and Poly_{3}(a, 0, c, 0, z) = 0 and 4 · a · c 0, then z = \frac{\sqrt{− 4·a·c}}{2·a}or z = \frac{− \sqrt{− 4·a·c}}{2·a}or z = 0.

(18) If a ≠ 0 and Poly_{3}(a, b, c, 0, z) = 0 and Δ(a, b, c) 0, then z = \frac{− b+\sqrt{Δ(a,b,c)}}{2·a} or z = ^{\frac{− b−\sqrt{Δ(a,b,c)}}{2·a}} or z = − \frac{b}{2·a} or z = 0.

(19) If a ≠ 0 and Poly_{3}(a, b, c, 0, z) = 0_{ℂ} and Δ(a, b, c) < 0, then z = − \frac{b}{2·a}+ \frac{\sqrt{− Δ(a,b,c)}}{2·a}i or z = − \frac{b}{2·a}+ (− ^{\frac{\sqrt{− Δ(a,b,c)}}{2·a}})i or z = 0_{ℂ}.

solving roots of polynomial equation … 87

(20) If a 0 and y^{2} = a, then y = \sqrt{a} or y = − \sqrt{a}.

(21) Suppose a ≠ 0 and Poly_{3}(a, 0, c, d, z) = 0_{ℂ} and ℑ(z) = 0. Let given u, v.

Suppose ℜ(z) = u + v and 3 · v · u + \frac{c}{a} = 0. Then

(i) z = ^{3} \sqrt{− \frac{d}{2·a}+ \sqrt{\frac{d^{2}}{4·a^{2}} + (\frac{c}{3·a})^{3}}} + ^{3} \sqrt{− \frac{d}{2·a}− \sqrt{\frac{d^{2}}{4·a^{2}} + (\frac{c}{3·a})^{3}}} , or (ii) z = ^{3} \sqrt{− \frac{d}{2·a}+ \sqrt{\frac{d^{2}}{4·a^{2}} + (\frac{c}{3·a})^{3}}} + ^{3} \sqrt{− \frac{d}{2·a}+ \sqrt{\frac{d^{2}}{4·a^{2}} + (\frac{c}{3·a})^{3}}}, or (iii) z = ^{3} \sqrt{− \frac{d}{2·a}− \sqrt{\frac{d^{2}}{4·a^{2}} + (\frac{c}{3·a})^{3}}} + ^{3} \sqrt{− \frac{d}{2·a}− \sqrt{\frac{d^{2}}{4·a^{2}} + (\frac{c}{3·a})^{3}}} .

(22) Suppose a ≠ 0 and Poly_{3}(a, 0, c, d, z) = 0_{ℂ} and ℑ(z) ≠ 0. Let given u, v.

Suppose ℜ(z) = u + v and 3 · v · u + \frac{c}{4·a} = 0 and \frac{c}{a} 0. Then

(i) z = ( ^{3} \sqrt{\frac{d}{16·a}+ \sqrt{(\frac{d}{16·a})^{2} + (\frac{c}{12·a})^{3}}} + ^{3} \sqrt{\frac{d}{16·a}− \sqrt{(\frac{d}{16·a})^{2} + (\frac{c}{12·a})^{3}}} ) + \sqrt{3 · ( ^{3} \sqrt{\frac{d}{16·a}+ \sqrt{(\frac{d}{16·a})^{2} + (\frac{c}{12·a})^{3}}} + ^{3} \sqrt{\frac{d}{16·a}− \sqrt{(\frac{d}{16·a})^{2} + (\frac{c}{12·a})^{3}}} )^{2} + \frac{c}{a}}i, or

(ii) z = ( ^{3} \sqrt{\frac{d}{16·a}+ \sqrt{(\frac{d}{16·a})^{2} + (\frac{c}{12·a})^{3}}} + ^{3} \sqrt{\frac{d}{16·a}− \sqrt{(\frac{d}{16·a})^{2} + (\frac{c}{12·a})^{3}}} ) + (− \sqrt{3 · ( ^{3} \sqrt{\frac{d}{16·a}+ \sqrt{(\frac{d}{16·a})^{2} + (\frac{c}{12·a})^{3}}} + ^{3} \sqrt{\frac{d}{16·a}− \sqrt{(\frac{d}{16·a})^{2} + (\frac{c}{12·a})^{3}}} )^{2} + \frac{c}{a}})i, or

(iii) z = 2 · ^{3} \sqrt{\frac{d}{16·a}+ \sqrt{(\frac{d}{16·a})^{2} + (\frac{c}{12·a})^{3}}}+

\sqrt{3 · (2 · ^{3} \sqrt{\frac{d}{16·a}+ \sqrt{(\frac{d}{16·a})^{2} + (\frac{c}{12·a})^{3}}})^{2} + \frac{c}{a}}i, or (iv) z = 2 · ^{3} \sqrt{\frac{d}{16·a}+ \sqrt{(\frac{d}{16·a})^{2} + (\frac{c}{12·a})^{3}}}+

(− \sqrt{3 · (2 · ^{3} \sqrt{\frac{d}{16·a}+ \sqrt{(\frac{d}{16·a})^{2} + (\frac{c}{12·a})^{3}}})^{2} + \frac{c}{a}})i, or

(v) z = 2 · ^{3} \sqrt{\frac{d}{16·a}− \sqrt{(\frac{d}{16·a})^{2} + (\frac{c}{12·a})^{3}}} + \sqrt{3 · (2 · ^{3} \sqrt{\frac{d}{16·a}− \sqrt{(\frac{d}{16·a})^{2} + (\frac{c}{12·a})^{3}}} )^{2} + \frac{c}{a}}i, or

(vi) z = 2 · ^{3} \sqrt{\frac{d}{16·a}− \sqrt{(\frac{d}{16·a})^{2} + (\frac{c}{12·a})^{3}}} +

(− \sqrt{3 · (2 · ^{3} \sqrt{\frac{d}{16·a}− \sqrt{(\frac{d}{16·a})^{2} + (\frac{c}{12·a})^{3}}} )^{2} + \frac{c}{a}})i.

(23) Suppose a ≠ 0 and Poly_{3}(a, b, c, d, z) = 0_{ℂ} and ℑ(z) = 0. Let given u, v, x_{1}. Suppose x_{1} = ℜ(z)+\frac{b}{3·a}and ℜ(z) = (u+v)−\frac{b}{3·a}and 3·u·v+\frac{3·a·c−b^{2}}{3·a^{2}} = 0. Then (i) z = (( ^{3} \sqrt{(− (\frac{b}{3·a})^{3} − \frac{3·a·d−b·c}{6·a^{2}}) + \sqrt{\frac{(2·( \frac{b}{3·a})^{3}+ \frac{3·a·d−b·c}{3·a^{2}} )^{2}}{4} + (\frac{3·a·c−b^{2}}{9·a^{2}} )^{3}}} +

3 \sqrt{− (\frac{b}{3·a})^{3} − \frac{3·a·d−b·c}{6·a^{2}}− \sqrt{\frac{(2·( \frac{b}{3·a})^{3}+ \frac{3·a·d−b·c}{3·a^{2}} )^{2}}{4} + (\frac{3·a·c−b^{2}}{9·a^{2}} )^{3}}} ) − \frac{b}{3·a}) + 0i, or

(ii) z = (( ^{3} \sqrt{(− (\frac{b}{3·a})^{3} − \frac{3·a·d−b·c}{6·a^{2}}) + \sqrt{\frac{(2·( \frac{b}{3·a})^{3}+ \frac{3·a·d−b·c}{3·a^{2}} )^{2}}{4} + (\frac{3·a·c−b^{2}}{9·a^{2}} )^{3}}} +

3 \sqrt{(− (\frac{b}{3·a})^{3} − \frac{3·a·d−b·c}{6·a^{2}}) + \sqrt{\frac{(2·( \frac{b}{3·a})^{3}+ \frac{3·a·d−b·c}{3·a^{2}} )^{2}}{4} + (\frac{3·a·c−b^{2}}{9·a^{2}} )^{3}}} ) − \frac{b}{3·a}) + 0i, or

(iii) z = (( ^{3} \sqrt{− (\frac{b}{3·a})^{3} − \frac{3·a·d−b·c}{6·a^{2}}− \sqrt{\frac{(2·( \frac{b}{3·a})^{3}+ \frac{3·a·d−b·c}{3·a^{2}} )^{2}}{4} + (\frac{3·a·c−b^{2}}{9·a^{2}} )^{3}}} +

3 \sqrt{− (\frac{b}{3·a})^{3} − \frac{3·a·d−b·c}{6·a^{2}}− \sqrt{\frac{(2·( \frac{b}{3·a})^{3}+ \frac{3·a·d−b·c}{3·a^{2}} )^{2}}{4} + (\frac{3·a·c−b^{2}}{9·a^{2}} )^{3}}} ) − \frac{b}{3·a}) + 0i.

(24) If z_{1} ≠ 0 and Poly1(z_{1}, z_{2}, z) = 0, then z = − \frac{z_{2}}{z_{1}} .

(25) If z_{2} ≠ 0, then it is not true that there exists z such that Poly1(0, z_{2}, z) = 0.

2. Complex Roots of Polynomial Equation of Degree 2 and 3 with Complex Coefficients

Let us consider z_{1}, z_{2}, z_{3}, z. The functor CPoly2(z_{1}, z_{2}, z_{3}, z) yields an element of ℂ and is defined by:

(Def. 6) CPoly2(z_{1}, z_{2}, z_{3}, z) = z_{1} · z^{2} + z_{2} · z + z_{3}.

We now state a number of propositions:

(26) If for every z holds CPoly2(z_{1}, z_{2}, z_{3}, z) = CPoly2(s_{1}, s_{2}, s_{3}, z), then z_{1} = s_{1} and z_{2} = s_{2} and z_{3} = s_{3}.

(27) \frac{− a+\sqrt{a^{2} +b^{2}}}{2} 0 and \frac{a+\sqrt{a^{2} +b^{2}}}{2} 0.

(28) If z^{2} = s and ℑ(s) 0, then z = \sqrt{\frac{ℜ(s)+\sqrt{ℜ(s)^{2} +ℑ(s)^{2}}}{2}} + \sqrt{\frac{− ℜ(s) +\sqrt{ℜ(s)^{2} +ℑ(s)^{2}}}{2}} i or z =

− \sqrt{\frac{ℜ(s)+\sqrt{ℜ(s)^{2} +ℑ(s)^{2}}}{2}} + (− \sqrt{\frac{− ℜ(s) +\sqrt{ℜ(s)^{2} +ℑ(s)^{2}}}{2}} )i.

(29) If z^{2} = s and ℑ(s) = 0 and ℜ(s) > 0, then z = \sqrt{ℜ(s)} or z = − \sqrt{ℜ(s)} .

(30) If z^{2} = s and ℑ(s) = 0 and ℜ(s) < 0, then z = 0 + \sqrt{− ℜ(s)} i or z = 0 + (− \sqrt{− ℜ(s)} )i.

(31) If z^{2} = s and ℑ(s) < 0, then z = \sqrt{\frac{ℜ(s)+\sqrt{ℜ(s)^{2} +ℑ(s)^{2}}}{2}} + (− \sqrt{\frac{− ℜ(s) +\sqrt{ℜ(s)^{2} +ℑ(s)^{2}}}{2}} )i or z =

− \sqrt{\frac{ℜ(s)+\sqrt{ℜ(s)^{2} +ℑ(s)^{2}}}{2}} + \sqrt{\frac{− ℜ(s) +\sqrt{ℜ(s)^{2} +ℑ(s)^{2}}}{2}} i.

solving roots of polynomial equation … 89

(32) Suppose z^{2} = s. Then (i) z = \sqrt{\frac{ℜ(s)+\sqrt{ℜ(s)^{2} +ℑ(s)^{2}}}{2}} + \sqrt{\frac{− ℜ(s) +\sqrt{ℜ(s)^{2} +ℑ(s)^{2}}}{2}} i, or (ii) z = − \sqrt{\frac{ℜ(s)+\sqrt{ℜ(s)^{2} +ℑ(s)^{2}}}{2}} + (− \sqrt{\frac{− ℜ(s) +\sqrt{ℜ(s)^{2} +ℑ(s)^{2}}}{2}} )i, or (iii) z = \sqrt{\frac{ℜ(s)+\sqrt{ℜ(s)^{2} +ℑ(s)^{2}}}{2}} + (− \sqrt{\frac{− ℜ(s) +\sqrt{ℜ(s)^{2} +ℑ(s)^{2}}}{2}} )i, or (iv) z = − \sqrt{\frac{ℜ(s)+\sqrt{ℜ(s)^{2} +ℑ(s)^{2}}}{2}} + \sqrt{\frac{− ℜ(s) +\sqrt{ℜ(s)^{2} +ℑ(s)^{2}}}{2}} i.

(33) CPoly2(0_{ℂ}, 0_{ℂ}, 0_{ℂ}, z) = 0.

(34) If z_{1} ≠ 0 and CPoly2(z_{1}, 0_{ℂ}, 0_{ℂ}, z) = 0, then z = 0.

(35) If z_{1} ≠ 0 and CPoly2(z_{1}, z_{2}, 0_{ℂ}, z) = 0, then z = − \frac{z_{2}}{z_{1}} or z = 0.

(36) Suppose z_{1} ≠ 0_{ℂ} and CPoly2(z_{1}, 0_{ℂ}, z_{3}, z) = 0_{ℂ}. Let given s. Suppose

s = − \frac{z_{3}}{z_{1}} . Then

(i) z = \sqrt{\frac{ℜ(s)+\sqrt{ℜ(s)^{2} +ℑ(s)^{2}}}{2}} + \sqrt{\frac{− ℜ(s) +\sqrt{ℜ(s)^{2} +ℑ(s)^{2}}}{2}} i, or (ii) z = − \sqrt{\frac{ℜ(s)+\sqrt{ℜ(s)^{2} +ℑ(s)^{2}}}{2}} + (− \sqrt{\frac{− ℜ(s) +\sqrt{ℜ(s)^{2} +ℑ(s)^{2}}}{2}} )i, or (iii) z = \sqrt{\frac{ℜ(s)+\sqrt{ℜ(s)^{2} +ℑ(s)^{2}}}{2}} + (− \sqrt{\frac{− ℜ(s) +\sqrt{ℜ(s)^{2} +ℑ(s)^{2}}}{2}} )i, or (iv) z = − \sqrt{\frac{ℜ(s)+\sqrt{ℜ(s)^{2} +ℑ(s)^{2}}}{2}} + \sqrt{\frac{− ℜ(s) +\sqrt{ℜ(s)^{2} +ℑ(s)^{2}}}{2}} i.

(37) Suppose z_{1} ≠ 0 and CPoly2(z_{1}, z_{2}, z_{3}, z) = 0_{ℂ}. Let given h, t. Suppose

h = (\frac{z_{2}}{2·z_{1}})^{2} − \frac{z_{3}}{z_{1}} and t = \frac{z_{2}}{2·z_{1}}. Then

(i) z = (\sqrt{\frac{ℜ(h)+\sqrt{ℜ(h)^{2} +ℑ(h)^{2}}}{2}} + \sqrt{\frac{− ℜ(h) +\sqrt{ℜ(h)^{2} +ℑ(h)^{2}}}{2}} i) − t, or (ii) z = (− \sqrt{\frac{ℜ(h)+\sqrt{ℜ(h)^{2} +ℑ(h)^{2}}}{2}} + (− \sqrt{\frac{− ℜ(h) +\sqrt{ℜ(h)^{2} +ℑ(h)^{2}}}{2}} )i) − t, or (iii) z = (\sqrt{\frac{ℜ(h)+\sqrt{ℜ(h)^{2} +ℑ(h)^{2}}}{2}} + (− \sqrt{\frac{− ℜ(h) +\sqrt{ℜ(h)^{2} +ℑ(h)^{2}}}{2}} )i) − t, or (iv) z = (− \sqrt{\frac{ℜ(h)+\sqrt{ℜ(h)^{2} +ℑ(h)^{2}}}{2}} + \sqrt{\frac{− ℜ(h) +\sqrt{ℜ(h)^{2} +ℑ(h)^{2}}}{2}} i) − t. Let us consider z_{1}, z_{2}, z_{3}, z_{4}, z. The functor CPoly2(z_{1}, z_{2}, z_{3}, z_{4}, z) yields

an element of ℂ and is defined as follows: (Def. 7) CPoly2(z_{1}, z_{2}, z_{3}, z_{4}, z) = z_{1} · z^{3} + z_{2} · z^{2} + z_{3} · z + z_{4}.

One can prove the following propositions:

(38) If z^{2} = 1, then z = 1 or z = − 1.

(39) z_{ℕ}^{3} = z · z · z and z_{ℕ}^{3} = z^{2} · z and z_{ℕ}^{3} = z^{3}.

(40) If z_{1} ≠ 0 and CPoly2(z_{1}, z_{2}, 0_{ℂ}, 0_{ℂ}, z) = 0_{ℂ}, then z = − \frac{z_{2}}{z_{1}} or z = 0.

(41) Suppose z_{1} ≠ 0_{ℂ} and CPoly2(z_{1}, 0_{ℂ}, z_{3}, 0_{ℂ}, z) = 0_{ℂ}. Let given s. Suppose

s = − \frac{z_{3}}{z_{1}} . Then

(i) z = 0_{ℂ}, or (ii) z = \sqrt{\frac{ℜ(s)+\sqrt{ℜ(s)^{2} +ℑ(s)^{2}}}{2}} + \sqrt{\frac{− ℜ(s) +\sqrt{ℜ(s)^{2} +ℑ(s)^{2}}}{2}} i, or (iii) z = − \sqrt{\frac{ℜ(s)+\sqrt{ℜ(s)^{2} +ℑ(s)^{2}}}{2}} + (− \sqrt{\frac{− ℜ(s) +\sqrt{ℜ(s)^{2} +ℑ(s)^{2}}}{2}} )i, or (iv) z = \sqrt{\frac{ℜ(s)+\sqrt{ℜ(s)^{2} +ℑ(s)^{2}}}{2}} + (− \sqrt{\frac{− ℜ(s) +\sqrt{ℜ(s)^{2} +ℑ(s)^{2}}}{2}} )i, or

(v) z = − \sqrt{\frac{ℜ(s)+\sqrt{ℜ(s)^{2} +ℑ(s)^{2}}}{2}} + \sqrt{\frac{− ℜ(s) +\sqrt{ℜ(s)^{2} +ℑ(s)^{2}}}{2}} i.

(42) Suppose z_{1} ≠ 0 and CPoly2(z_{1}, z_{2}, z_{3}, 0_{ℂ}, z) = 0_{ℂ}. Let given s, h, t.

Suppose s = − \frac{z_{3}}{z_{1}} and h = (\frac{z_{2}}{2·z_{1}})^{2} − \frac{z_{3}}{z_{1}} and t = \frac{z_{2}}{2·z_{1}}. Then

(i) z = 0, or (ii) z = (\sqrt{\frac{ℜ(h)+\sqrt{ℜ(h)^{2} +ℑ(h)^{2}}}{2}} + \sqrt{\frac{− ℜ(h) +\sqrt{ℜ(h)^{2} +ℑ(h)^{2}}}{2}} i) − t, or

(iii) z = (− \sqrt{\frac{ℜ(h)+\sqrt{ℜ(h)^{2} +ℑ(h)^{2}}}{2}} + (− \sqrt{\frac{− ℜ(h) +\sqrt{ℜ(h)^{2} +ℑ(h)^{2}}}{2}} )i) − t, or

(iv) z = (\sqrt{\frac{ℜ(h)+\sqrt{ℜ(h)^{2} +ℑ(h)^{2}}}{2}} + (− \sqrt{\frac{− ℜ(h) +\sqrt{ℜ(h)^{2} +ℑ(h)^{2}}}{2}} )i) − t, or

(v) z = (− \sqrt{\frac{ℜ(h)+\sqrt{ℜ(h)^{2} +ℑ(h)^{2}}}{2}} + \sqrt{\frac{− ℜ(h) +\sqrt{ℜ(h)^{2} +ℑ(h)^{2}}}{2}} i) − t.

(43) If z = s − (\frac{1}{3} + 0i) · z_{1}, then z^{2} = s^{2} + (− (\frac{2}{3} + 0i)) · z_{1} · s + (\frac{1}{9} + 0i) · z_{1}^{2}.

(44) If z = s−(\frac{1}{3}+0i)·z_{1}, then z^{3} = ((s^{3}−z_{1}·s^{2})+(\frac{1}{3}+0i)·z_{1}^{2}·s)−(\frac{1}{27}+0i)·z_{1}^{3}.

(45) Suppose CPoly2(1_{ℂ}, z_{1}, z_{2}, z_{3}, z) = 0_{ℂ}. Let given p, q, s. Suppose z = s − (\frac{1}{3} + 0i) · z_{1} and p = − (\frac{1}{3} + 0i) · z_{1}^{2}+ z_{2} and q = ((\frac{2}{27} + 0i) · z_{1}^{3} − (\frac{1}{3} + 0i) · z_{1} · z_{2}) + z_{3}. Then CPoly2(1_{ℂ}, 0_{ℂ}, p, q, s) = 0_{ℂ}.

(46) For every element z of ℂ holds |z| · cosArgz + (|z| · sinArgz)i = (|z| + 0i) · (cosArgz + sinArgzi).

(47) For every element z of ℂ and for every natural number n holds z_{ℕ}^{n+1} = (z_{ℕ}^{n}) · z.

(48) For every element z of ℂ holds z_{ℕ}^{1} = z.

(49) For every element z of ℂ holds z_{ℕ}^{2} = z · z.

(50) For every natural number n such that n > 0 holds 0_{ℕ}^{n} = 0.

(51) For all elements x, y of ℂ and for every natural number n holds (x·y)_{ℕ}^{n} = (x_{ℕ}^{n}) · y_{ℕ}^{n}.

(52) For every real number x such that x > 0 and for every natural number n holds (x + 0i)_{ℕ}^{n} = x^{n} + 0i.

(53) For every real number x and for every natural number n holds (cosx + sinxi)_{ℕ}^{n} = cos(n · x) + sin(n · x)i.

(54) For every element z of ℂ and for every natural number n such that z ≠ 0_{ℂ} or n > 0 holds z_{ℕ}^{n} = |z|^{n} · cos(n · Argz) + (|z|^{n} · sin(n · Argz))i.

(55) For all natural numbers n, k and for every real number x such that n ≠ 0 holds (cos(\frac{x+2·π·k}{n}) + sin(\frac{x+2·π·k}{n})i)_{ℕ}^{n} = cosx + sinxi.

(56) Let z be an element of ℂ and n, k be natural numbers. If n ≠ 0, then

z = ( ^{n} \sqrt{|z|} · cos(\frac{Arg z+2·π·k}{n}) + ( ^{n} \sqrt{|z|} · sin(\frac{Arg z+2·π·k}{n}))i)_{ℕ}^{n}. Let z be an element of ℂ and let n be a non empty natural number. An

element of ℂ is called a complex root of n, z if: (Def. 8) It_{ℕ}^{n} = z.

solving roots of polynomial equation … 91

Next we state several propositions:

(57) Let z be an element of ℂ, n be a non empty natural number, and k be a natural number. Then ^{n} \sqrt{|z|} · cos(\frac{Arg z+2·π·k}{n}) + ( ^{n} \sqrt{|z|} · sin(\frac{Arg z+2·π·k}{n}))i is a complex root of n, z.

(58) For every element z of ℂ and for every complex root v of 1, z holds v = z.

(59) For every non empty natural number n and for every complex root v of n, 0_{ℂ} holds v = 0_{ℂ}.

(60) Let n be a non empty natural number, z be an element of ℂ, and v be a complex root of n, z. If v = 0_{ℂ}, then z = 0_{ℂ}.

(61) Let n be a non empty natural number and k be a natural number. Then cos(\frac{2·π·k}{n}) + sin(\frac{2·π·k}{n})i is a complex root of n, 1_{ℂ}.

(62) For every natural number k holds cos(\frac{2·π·k}{3}) + sin(\frac{2·π·k}{3})i is a complex root of 3, 1_{ℂ}.

(63) For all elements z, s of ℂ and for every natural number n such that s ≠ 0 and z ≠ 0 and n 1 and s_{ℕ}^{n} = z_{ℕ}^{n} holds |s| = |z|.

References

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92 yuzhong ding and xiquan liang

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Complex Linear Space and Complex Normed Space

Noboru Endou Gifu National College of Technology

Summary. In this article, we introduce the notion of complex linear space and complex normed space.

MML Identifier: CLVECT 1.

The articles [16], [7], [18], [1], [14], [13], [15], [8], [19], [4], [5], [2], [11], [17], [6], [10], [9], [3], and [12] provide the terminology and notation for this paper.

1. Complex Linear Space

We consider CLS structures as extensions of loop structure as systems ⟨ a carrier, a zero, an addition, an external multiplication ⟩,

where the carrier is a set, the zero is an element of the carrier, the addition is a binary operation on the carrier, and the external multiplication is a function from [: ℂ, the carrier :] into the carrier.

Let us observe that there exists a CLS structure which is non empty.

Let V be a CLS structure. A vector of V is an element of V.

Let V be a non empty CLS structure, let v be a vector of V, and let z be a Complex. The functor z · v yielding an element of V is defined as follows:

(Def. 1) z · v = (the external multiplication of V)(⟨⟨z, v⟩⟩).

Let Z_{1} be a non empty set, let O be an element of Z_{1}, let F be a binary operation on Z_{1}, and let G be a function from [: ℂ, Z_{1} :] into Z_{1}. One can verify that ⟨Z_{1}, O, F, G⟩ is non empty.

Let I_{1} be a non empty CLS structure. We say that I_{1} is complex linear space-like if and only if the conditions (Def. 2) are satisfied.

(Def. 2)(i) For every Complex z and for all vectors v, w of I_{1} holds z · (v + w) =

z · v + z · w, (ii) for all Complexes z_{1}, z_{2} and for every vector v of I_{1} holds (z_{1} +z_{2})·v =

z_{1} · v + z_{2} · v, (iii) for all Complexes z_{1}, z_{2} and for every vector v of I_{1} holds (z_{1} · z_{2}) · v =

z_{1} · (z_{2} · v), and (iv) for every vector v of I_{1} holds 1_{ℂ} · v = v. Let us observe that there exists a non empty CLS structure which is non

empty, strict, Abelian, add-associative, right zeroed, right complementable, and complex linear space-like.

A complex linear space is an Abelian add-associative right zeroed right complementable complex linear space-like non empty CLS structure.

One can prove the following proposition

(1) Let V be a non empty CLS structure. Suppose that for all vectors v, w of V holds v + w = w + v and for all vectors u, v, w of V holds (u + v) + w = u + (v + w) and for every vector v of V holds v + 0_{V} = v and for every vector v of V there exists a vector w of V such that v + w = 0_{V} and for every Complex z and for all vectors v, w of V holds z · (v + w) = z · v + z · w and for all Complexes z_{1}, z_{2} and for every vector v of V holds (z_{1} + z_{2}) · v = z_{1} · v + z_{2} · v and for all Complexes z_{1}, z_{2} and for every vector v of V holds (z_{1} · z_{2}) · v = z_{1} · (z_{2} · v) and for every vector v of V holds 1_{ℂ} · v = v. Then V is a complex linear space.

We adopt the following convention: V, X, Y are complex linear spaces, u, v, v_{1}, v_{2} are vectors of V, and z, z_{1}, z_{2} are Complexes.

The following propositions are true:

(2) If z = 0_{ℂ} or v = 0_{V} , then z · v = 0_{V} .

(3) If z · v = 0_{V} , then z = 0_{ℂ} or v = 0_{V} .

(4) − v = (− 1_{ℂ}) · v.

(5) If v = − v, then v = 0_{V} .

(6) If v + v = 0_{V} , then v = 0_{V} .

(7) z · − v = (− z) · v.

(8) z · − v = − z · v.

(9) (− z) · − v = z · v.

(10) z · (v − u) = z · v − z · u.

(11) (z_{1} − z_{2}) · v = z_{1} · v − z_{2} · v.

(12) If z ≠ 0 and z · v = z · u, then v = u.

(13) If v ≠ 0_{V} and z_{1} · v = z_{2} · v, then z_{1} = z_{2}.

(14) Let F, G be finite sequences of elements of the carrier of V. Suppose lenF = lenG and for every natural number k and for every vector v of V

such that k ∈ domF and v = G(k) holds F (k) = z·v. Then ∑ F = z·∑ G.

(15) z · ∑(ε_{(the carrier of V)}) = 0_{V} .

(16) z · ∑⟨v, u⟩ = z · v + z · u.

(17) z · ∑⟨u, v_{1}, v_{2}⟩ = z · u + z · v_{1} + z · v_{2}.

(18) ∑⟨v, v⟩ = (2 + 0i) · v.

(19) ∑⟨− v, − v⟩ = (− 2 + 0i) · v.

(20) ∑⟨v, v, v⟩ = (3 + 0i) · v.

2. Subspace and Cosets of Subspaces in Complex Linear Space

In the sequel V_{1}, V_{2}, V_{3} are subsets of V.

Let us consider V, V_{1}. We say that V_{1} is linearly closed if and only if the conditions (Def. 3) are satisfied.

(Def. 3)(i) For all vectors v, u of V such that v ∈ V_{1} and u ∈ V_{1} holds v +u ∈ V_{1},

and (ii) for every Complex z and for every vector v of V such that v ∈ V_{1} holds

z · v ∈ V_{1}. Next we state several propositions:

(21) If V_{1} ≠ ∅ and V_{1} is linearly closed, then 0_{V} ∈ V_{1}.

(22) If V_{1} is linearly closed, then for every vector v of V such that v ∈ V_{1} holds − v ∈ V_{1}.

(23) If V_{1} is linearly closed, then for all vectors v, u of V such that v ∈ V_{1} and u ∈ V_{1} holds v − u ∈ V_{1}.

(24) {0_{V} } is linearly closed.

(25) If the carrier of V = V_{1}, then V_{1} is linearly closed.

(26) If V_{1} is linearly closed and V_{2} is linearly closed and V_{3} = {v + u : v ∈ V_{1} ∧ u ∈ V_{2}}, then V_{3} is linearly closed.

(27) If V_{1} is linearly closed and V_{2} is linearly closed, then V_{1} ∩ V_{2} is linearly

closed. Let us consider V. A complex linear space is said to be a subspace of V if it

satisfies the conditions (Def. 4). (Def. 4)(i) The carrier of it ⊆ the carrier of V,

(ii) the zero of it = the zero of V, (iii) the addition of it = (the addition of V)↾[: the carrier of it, the carrier

of it :], and (iv) the external multiplication of it = (the external multiplication of

V)↾[: ℂ, the carrier of it :]. We use the following convention: W, W_{1}, W_{2} denote subspaces of V, x

denotes a set, and w, w_{1}, w_{2} denote vectors of W.

We now state a number of propositions:

(28) If x ∈ W_{1} and W_{1} is a subspace of W_{2}, then x ∈ W_{2}.

(29) If x ∈ W, then x ∈ V.

(30) w is a vector of V.

(31) 0_{W} = 0_{V} .

(32) 0_{(W_{1})} = 0_{(W_{2})}.

(33) If w_{1} = v and w_{2} = u, then w_{1} + w_{2} = v + u.

(34) If w = v, then z · w = z · v.

(35) If w = v, then − v = − w.

(36) If w_{1} = v and w_{2} = u, then w_{1} − w_{2} = v − u.

(37) 0_{V} ∈ W.

(38) 0_{(W_{1})} ∈ W_{2}.

(39) 0_{W} ∈ V.

(40) If u ∈ W and v ∈ W, then u + v ∈ W.

(41) If v ∈ W, then z · v ∈ W.

(42) If v ∈ W, then − v ∈ W.

(43) If u ∈ W and v ∈ W, then u − v ∈ W.

In the sequel D denotes a non empty set, d_{1} denotes an element of D, A denotes a binary operation on D, and M denotes a function from [: ℂ, D:] into D.

Next we state several propositions:

(44) Suppose V_{1} = D and d_{1} = 0_{V} and A = (the addition of V)↾[: V_{1}, V_{1} :] and M = (the external multiplication of V)↾[: ℂ, V_{1} :]. Then ⟨D, d_{1}, A, M⟩ is a subspace of V.

(45) V is a subspace of V.

(46) Let V, X be strict complex linear spaces. If V is a subspace of X and X is a subspace of V, then V = X.

(47) If V is a subspace of X and X is a subspace of Y, then V is a subspace of Y.

(48) If the carrier of W_{1} ⊆ the carrier of W_{2}, then W_{1} is a subspace of W_{2}.

(49) If for every v such that v ∈ W_{1} holds v ∈ W_{2}, then W_{1} is a subspace of

W_{2}. Let us consider V. Observe that there exists a subspace of V which is strict. The following propositions are true:

(50) For all strict subspaces W_{1}, W_{2} of V such that the carrier of W_{1} = the carrier of W_{2} holds W_{1} = W_{2}.

(51) For all strict subspaces W_{1}, W_{2} of V such that for every v holds v ∈ W_{1} iff v ∈ W_{2} holds W_{1} = W_{2}.

(52) Let V be a strict complex linear space and W be a strict subspace of V. If the carrier of W = the carrier of V, then W = V.

(53) Let V be a strict complex linear space and W be a strict subspace of V. If for every vector v of V holds v ∈ W iff v ∈ V, then W = V.

(54) If the carrier of W = V_{1}, then V_{1} is linearly closed.

(55) If V_{1} ≠ ∅ and V_{1} is linearly closed, then there exists a strict subspace W

of V such that V_{1} = the carrier of W. Let us consider V. The functor 0_{V} yields a strict subspace of V and is defined

by: (Def. 5) The carrier of 0_{V} = {0_{V} }.

Let us consider V. The functor Ω_{V} yields a strict subspace of V and is defined as follows:

(Def. 6) Ω_{V} = the CLS structure of V.

We now state several propositions:

(56) 0_{W} = 0_{V} .

(57) 0_{(W_{1})} = 0_{(W_{2})}.

(58) 0_{W} is a subspace of V.

(59) 0_{V} is a subspace of W.

(60) 0_{(W_{1})} is a subspace of W_{2}.

(61) Every strict complex linear space V is a subspace of Ω_{V}.

Let us consider V and let us consider v, W. The functor v + W yielding a subset of V is defined by:

(Def. 7) v + W = {v + u : u ∈ W }.

Let us consider V and let us consider W. A subset of V is called a coset of W if:

(Def. 8) There exists v such that it = v + W. In the sequel B, C denote cosets of W. The following propositions are true:

(62) 0_{V} ∈ v + W iff v ∈ W.

(63) v ∈ v + W.

(64) 0_{V} + W = the carrier of W.

(65) v + 0_{V} = {v}.

(66) v + Ω_{V} = the carrier of V.

(67) 0_{V} ∈ v + W iff v + W = the carrier of W.

(68) v ∈ W iff v + W = the carrier of W.

(69) If v ∈ W, then z · v + W = the carrier of W.

(70) If z ≠ 0_{ℂ} and z · v + W = the carrier of W, then v ∈ W.

(71) v ∈ W iff − v + W = the carrier of W.

(72) u ∈ W iff v + W = v + u + W.

(73) u ∈ W iff v + W = (v − u) + W.

(74) v ∈ u + W iff u + W = v + W.

(75) v + W = − v + W iff v ∈ W.

(76) If u ∈ v_{1} + W and u ∈ v_{2} + W, then v_{1} + W = v_{2} + W.

(77) If u ∈ v + W and u ∈ − v + W, then v ∈ W.

(78) If z ≠ 1_{ℂ} and z · v ∈ v + W, then v ∈ W.

(79) If v ∈ W, then z · v ∈ v + W.

(80) − v ∈ v + W iff v ∈ W.

(81) u + v ∈ v + W iff u ∈ W.

(82) v − u ∈ v + W iff u ∈ W.

(83) u ∈ v + W iff there exists v_{1} such that v_{1} ∈ W and u = v + v_{1}.

(84) u ∈ v + W iff there exists v_{1} such that v_{1} ∈ W and u = v − v_{1}.

(85) There exists v such that v_{1} ∈ v + W and v_{2} ∈ v + W iff v_{1} − v_{2} ∈ W.

(86) If v + W = u + W, then there exists v_{1} such that v_{1} ∈ W and v + v_{1} = u.

(87) If v + W = u + W, then there exists v_{1} such that v_{1} ∈ W and v − v_{1} = u.

(88) For all strict subspaces W_{1}, W_{2} of V holds v +W_{1} = v +W_{2} iff W_{1} = W_{2}.

(89) For all strict subspaces W_{1}, W_{2} of V such that v + W_{1} = u + W_{2} holds W_{1} = W_{2}.

(90) C is linearly closed iff C = the carrier of W.

(91) For all strict subspaces W_{1}, W_{2} of V and for every coset C_{1} of W_{1} and for every coset C_{2} of W_{2} such that C_{1} = C_{2} holds W_{1} = W_{2}.

(92) {v} is a coset of 0_{V} .

(93) If V_{1} is a coset of 0_{V} , then there exists v such that V_{1} = {v}.

(94) The carrier of W is a coset of W.

(95) The carrier of V is a coset of Ω_{V}.

(96) If V_{1} is a coset of Ω_{V}, then V_{1} = the carrier of V.

(97) 0_{V} ∈ C iff C = the carrier of W.

(98) u ∈ C iff C = u + W.

(99) If u ∈ C and v ∈ C, then there exists v_{1} such that v_{1} ∈ W and u+v_{1} = v.

(100) If u ∈ C and v ∈ C, then there exists v_{1} such that v_{1} ∈ W and u−v_{1} = v.

(101) There exists C such that v_{1} ∈ C and v_{2} ∈ C iff v_{1} − v_{2} ∈ W.

(102) If u ∈ B and u ∈ C, then B = C.

3. Complex Normed Space

We consider complex normed space structures as extensions of CLS structure as systems

⟨ a carrier, a zero, an addition, an external multiplication, a norm ⟩, where the carrier is a set, the zero is an element of the carrier, the addition is a binary operation on the carrier, the external multiplication is a function from [: ℂ, the carrier :] into the carrier, and the norm is a function from the carrier into ℝ.

Let us mention that there exists a complex normed space structure which is non empty.

In the sequel X is a non empty complex normed space structure and x is a point of X.

Let us consider X, x. The functor ∥x∥ yielding a real number is defined by: (Def. 9) ∥x∥ = (the norm of X)(x).

Let I_{1} be a non empty complex normed space structure. We say that I_{1} is complex normed space-like if and only if:

(Def. 10) For all points x, y of I_{1} and for every z holds ∥x∥ = 0 iff x = 0_{(I_{1})} and

∥z · x∥ = |z| · ∥x∥ and ∥x + y∥ ∥x∥ + ∥y∥. One can verify that there exists a non empty complex normed space structure

which is complex normed space-like, complex linear space-like, Abelian, addassociative, right zeroed, right complementable, and strict.

A complex normed space is a complex normed space-like complex linear space-like Abelian add-associative right zeroed right complementable non empty complex normed space structure.

We follow the rules: C_{3} is a complex normed space and x, y, w, g are points of C_{3}.

The following propositions are true:

(103) ∥0_{(C_{3})}∥ = 0.

(104) ∥− x∥ = ∥x∥.

(105) ∥x − y∥ ∥x∥ + ∥y∥.

(106) 0 ∥x∥.

(107) ∥z_{1} · x + z_{2} · y∥ |z_{1}| · ∥x∥ + |z_{2}| · ∥y∥.

(108) ∥x − y∥ = 0 iff x = y.

(109) ∥x − y∥ = ∥y − x∥.

(110) ∥x∥ − ∥y∥ ∥x − y∥.

(111) |∥x∥ − ∥y∥| ∥x − y∥.

(112) ∥x − w∥ ∥x − y∥ + ∥y − w∥.

(113) If x ≠ y, then ∥x − y∥ ≠ 0.

We adopt the following rules: S, S_{1}, S_{2} are sequences of C_{3}, n, m are natural numbers, and r is a real number.

One can prove the following proposition

(114) There exists S such that rngS = {0_{(C_{3})}}.

In this article we present several logical schemes. The scheme ExCNSSeq deals with a complex normed space A and a unary functor F yielding a point of A, and states that:

There exists a sequence S of A such that for every n holds S(n) = F(n)

for all values of the parameters.

The scheme ExCLSSeq deals with a complex linear space A and a unary functor F yielding an element of A, and states that:

There exists a sequence S of A such that for every n holds S(n) = F(n)

for all values of the parameters.

Let C_{3} be a complex linear space and let S_{1}, S_{2} be sequences of C_{3}. The functor S_{1} + S_{2} yielding a sequence of C_{3} is defined by:

(Def. 11) For every n holds (S_{1} + S_{2})(n) = S_{1}(n) + S_{2}(n).

Let C_{3} be a complex linear space and let S_{1}, S_{2} be sequences of C_{3}. The functor S_{1} − S_{2} yielding a sequence of C_{3} is defined by:

(Def. 12) For every n holds (S_{1} − S_{2})(n) = S_{1}(n) − S_{2}(n).

Let C_{3} be a complex linear space, let S be a sequence of C_{3}, and let x be an element of C_{3}. The functor S − x yielding a sequence of C_{3} is defined by:

(Def. 13) For every n holds (S − x)(n) = S(n) − x.

Let C_{3} be a complex linear space, let S be a sequence of C_{3}, and let us consider z. The functor z · S yields a sequence of C_{3} and is defined as follows:

(Def. 14) For every n holds (z · S)(n) = z · S(n).

Let us consider C_{3} and let us consider S. We say that S is convergent if and only if:

(Def. 15) There exists g such that for every r such that 0 < r there exists m such

that for every n such that m n holds ∥S(n) − g∥ < r. The following four propositions are true:

(115) If S_{1} is convergent and S_{2} is convergent, then S_{1} + S_{2} is convergent.

(116) If S_{1} is convergent and S_{2} is convergent, then S_{1} − S_{2} is convergent.

(117) If S is convergent, then S − x is convergent.

(118) If S is convergent, then z · S is convergent.

Let us consider C_{3} and let us consider S. The functor ∥S∥ yielding a sequence of real numbers is defined as follows:

(Def. 16) For every n holds ∥S∥(n) = ∥S(n)∥.

complex linear space and complex normed space 101

The following proposition is true

(119) If S is convergent, then ∥S∥ is convergent.

Let us consider C_{3} and let us consider S. Let us assume that S is convergent. The functor limS yields a point of C_{3} and is defined as follows:

(Def. 17) For every r such that 0 < r there exists m such that for every n such that m n holds ∥S(n) − limS∥ < r.

The following propositions are true:

(120) If S is convergent and limS = g, then ∥S −g∥ is convergent and lim∥S − g∥ = 0.

(121) If S_{1} is convergent and S_{2} is convergent, then lim(S_{1} + S_{2}) = limS_{1} + limS_{2}.

(122) If S_{1} is convergent and S_{2} is convergent, then lim(S_{1} − S_{2}) = limS_{1} − limS_{2}.

(123) If S is convergent, then lim(S − x) = limS − x.

(124) If S is convergent, then lim(z · S) = z · limS.

References

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[2] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107–114, 1990.

[3] Czesław Byliński. The complex numbers. Formalized Mathematics, 1(3):507–513, 1990.

[4] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55– 65, 1990.

[5] Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153–164, 1990.

[6] Czesław Byliński. Partial functions. Formalized Mathematics, 1(2):357–367, 1990.

[7] Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47–53, 1990.

[8] Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35–40, 1990.

[9] Jarosław Kotowicz. Convergent sequences and the limit of sequences. Formalized Mathematics, 1(2):273–275, 1990.

[10] Jarosław Kotowicz. Real sequences and basic operations on them. Formalized Mathematics, 1(2):269–272, 1990.

[11] Jan Popiołek. Some properties of functions modul and signum. Formalized Mathematics, 1(2):263–264, 1990.

[12] Jan Popiołek. Real normed space. Formalized Mathematics, 2(1):111–115, 1991.

[13] Andrzej Trybulec. Introduction to arithmetics. To appear in Formalized Mathematics.

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102 noboru endou

[17] Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291–296, 1990.

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Received January 26, 2004

The Banach Algebra of Bounded Linear Operators

Yasunari Shidama Shinshu University Nagano

Summary. In this article, the basic properties of Banach algebra are described. This algebra is defined as the set of all bounded linear operators from one normed space to another.

MML Identifier: LOPBAN 2.

The papers [21], [8], [23], [25], [24], [5], [7], [6], [19], [4], [1], [2], [18], [10], [22], [13], [3], [20], [16], [15], [9], [12], [11], [14], and [17] provide the terminology and notation for this paper.

Let X be a non empty set and let f, g be elements of X^{X}. Then g · f is an element of X^{X}.

One can prove the following propositions:

(1) Let X, Y, Z be real linear spaces, f be a linear operator from X into Y, and g be a linear operator from Y into Z. Then g · f is a linear operator from X into Z.

(2) Let X, Y, Z be real normed spaces, f be a bounded linear operator from X into Y, and g be a bounded linear operator from Y into Z. Then

(i) g · f is a bounded linear operator from X into Z, and (ii) for every vector x of X holds ∥(g ·f)(x)∥ (BdLinOpsNorm(Y, Z))(g)·

(BdLinOpsNorm(X, Y))(f) · ∥x∥ and (BdLinOpsNorm(X, Z))(g · f) (BdLinOpsNorm(Y, Z))(g) · (BdLinOpsNorm(X, Y))(f).

Let X be a real normed space and let f, g be bounded linear operators from X into X. Then g · f is a bounded linear operator from X into X.

Let X be a real normed space and let f, g be elements of BdLinOps(X, X). The functor f +g yields an element of BdLinOps(X, X) and is defined as follows:

(Def. 1) f + g = (Add (BdLinOps(X, X),

RVectorSpaceOfLinearOperators(X, X)))(f, g). Let X be a real normed space and let f, g be elements of BdLinOps(X, X).

The functor g · f yielding an element of BdLinOps(X, X) is defined as follows: (Def. 2) g · f = modetrans(g, X, X) · modetrans(f, X, X).

Let X be a real normed space, let f be an element of BdLinOps(X, X), and let a be a real number. The functor a · f yields an element of BdLinOps(X, X) and is defined by:

(Def. 3) a · f = (Mult (BdLinOps(X, X),

RVectorSpaceOfLinearOperators(X, X)))(a, f). Let X be a real normed space. The functor FuncMult(X) yielding a binary

operation on BdLinOps(X, X) is defined as follows: (Def. 4) For all elements f, g of BdLinOps(X, X) holds (FuncMult(X))(f, g) =

f · g. The following proposition is true

(3) For every real normed space X holds id_{the carrier of X} is a bounded linear operator from X into X.

Let X be a real normed space. The functor FuncUnit(X) yields an element of BdLinOps(X, X) and is defined as follows:

(Def. 5) FuncUnit(X) = id_{the carrier of X}.

One can prove the following propositions:

(4) Let X be a real normed space and f, g, h be bounded linear operators from X into X. Then h = f · g if and only if for every vector x of X holds h(x) = f(g(x)).

(5) For every real normed space X and for all bounded linear operators f, g, h from X into X holds f · (g · h) = (f · g) · h.

(6) Let X be a real normed space and f be a bounded linear operator from X into X. Then f · id_{the carrier of X} = f and id_{the carrier of X} · f = f.

(7) For every real normed space X and for all elements f, g, h of BdLinOps(X, X) holds f · (g · h) = (f · g) · h.

(8) For every real normed space X and for every element f of BdLinOps(X, X) holds f · FuncUnit(X) = f and FuncUnit(X) · f = f.

(9) For every real normed space X and for all elements f, g, h of BdLinOps(X, X) holds f · (g + h) = f · g + f · h.

(10) For every real normed space X and for all elements f, g, h of BdLinOps(X, X) holds (g + h) · f = g · f + h · f.

(11) Let X be a real normed space, f, g be elements of BdLinOps(X, X), and a, b be real numbers. Then (a · b) · (f · g) = a · f · (b · g).

(12) For every real normed space X and for all elements f, g of

BdLinOps(X, X) and for every real number a holds a · (f · g) = (a · f) · g. Let X be a real normed space. The functor RingOfBoundedLinearOperators(X)

yielding a double loop structure is defined as follows: (Def. 6) RingOfBoundedLinearOperators(X) = ⟨BdLinOps(X, X), Add (BdLinOps

(X, X), RVectorSpaceOfLinearOperators(X, X)), FuncMult(X), FuncUnit(X), Zero (BdLinOps(X, X), RVectorSpaceOfLinearOperators(X, X))⟩.

Let X be a real normed space. Observe that RingOfBoundedLinearOperators(X) is non empty and strict.

One can prove the following propositions:

(13) Let X be a real normed space and x, y, z be elements of RingOfBoundedLinearOperators(X). Then x + y = y + x and (x + y) + z = x + (y + z) and x + 0_{RingOfBoundedLinearOperators (X)} = x and there exists an element t of RingOfBoundedLinearOperators(X) such that x + t = 0_{RingOfBoundedLinearOperators (X)} and (x · y) · z = x · (y · z) and x · 1_{RingOfBoundedLinearOperators (X)} = x and 1_{RingOfBoundedLinearOperators (X)} · x = x and x · (y + z) = x · y + x · z and (y + z) · x = y · x + z · x.

(14) For every real normed space X holds RingOfBoundedLinearOperators(X)

is a ring. Let X be a real normed space. Note that RingOfBoundedLinearOperators(X)

is Abelian, add-associative, right zeroed, right complementable, associative, left unital, right unital, and distributive.

Let X be a real normed space. The functor RAlgebraOfBoundedLinearOperators(X) yielding an algebra

structure is defined as follows: (Def. 7) RAlgebraOfBoundedLinearOperators(X) = ⟨BdLinOps(X, X),

FuncMult(X), Add (BdLinOps(X, X), RVectorSpaceOfLinearOperators (X, X)), Mult (BdLinOps(X, X), RVectorSpaceOfLinearOperators(X, X)), FuncUnit(X), Zero (BdLinOps(X, X), RVectorSpaceOfLinearOperators (X, X))⟩.

Let X be a real normed space. Observe that RAlgebraOfBoundedLinearOperators(X) is non empty and

strict.

Next we state the proposition

(15) Let X be a real normed space, x, y, z be elements of RAlgebraOfBoundedLinearOperators(X), and a, b be real numbers. Then x + y = y + x and (x + y) + z = x + (y + z) and x + 0_{RAlgebraOfBoundedLinearOperators (X)} = x and there exists an element t of RAlgebraOfBoundedLinearOperators(X) such that x + t = 0_{RAlgebraOfBoundedLinearOperators (X)} and (x · y) · z = x · (y · z) and x · 1_{RAlgebraOfBoundedLinearOperators (X)} = x and 1_{RAlgebraOfBoundedLinearOperators (X)} · x = x and x · (y + z) = x · y + x · z and (y+z)·x = y·x+z·x and a·(x·y) = (a·x)·y and a·(x+y) = a·x+a·y and (a+b)·x = a·x+b·x and (a·b)·x = a·(b·x) and (a·b)·(x·y) = a·x·(b·y).

A BL algebra is an Abelian add-associative right zeroed right complementable associative algebra-like non empty algebra structure.

The following proposition is true

(16) For every real normed space X holds

RAlgebraOfBoundedLinearOperators(X) is a BL algebra. One can check that l1-Space is complete. Let us mention that l1-Space is non trivial. One can verify that there exists a real Banach space which is non trivial. One can prove the following propositions:

(17) For every non trivial real normed space X there exists a vector w of X such that ∥w∥ = 1.

(18) For every non trivial real normed space X holds (BdLinOpsNorm(X, X))

(id_{the carrier of X}) = 1. We introduce normed algebra structures which are extensions of algebra

structure and normed structure and are systems

⟨ a carrier, a multiplication, an addition, an external multiplication, a unity, a zero, a norm ⟩, where the carrier is a set, the multiplication and the addition are binary operations on the carrier, the external multiplication is a function from [: ℝ, the carrier :] into the carrier, the unity and the zero are elements of the carrier, and the norm is a function from the carrier into ℝ.

Let us mention that there exists a normed algebra structure which is non empty.

Let X be a real normed space. The functor RNormedAlgebraOfBoundedLinearOperators(X) yields a nor-

med algebra structure and is defined by: (Def. 8) RNormedAlgebraOfBoundedLinearOperators(X) = ⟨BdLinOps(X, X),

FuncMult(X), Add (BdLinOps(X, X), RVectorSpaceOfLinearOperators (X, X)), Mult (BdLinOps(X, X), RVectorSpaceOfLinearOperators(X, X)), FuncUnit(X), Zero (BdLinOps(X, X), RVectorSpaceOfLinearOperators (X, X)), BdLinOpsNorm(X, X)⟩.

Let X be a real normed space. One can verify that RNormedAlgebraOfBoundedLinearOperators(X) is non empty and strict. Next we state two propositions:

(19) Let X be a real normed space, x, y, z be elements of RNormedAlgebraOfBoundedLinearOperators(X), and a, b be real numbers. Then x + y = y + x and (x + y) + z = x + (y + z) and x + 0_{RNormedAlgebraOfBoundedLinearOperators (X)} = x and there exists an element t of RNormedAlgebraOfBoundedLinearOperators(X) such that x + t = 0_{RNormedAlgebraOfBoundedLinearOperators (X)} and (x · y) · z = x · (y · z) and x · 1_{RNormedAlgebraOfBoundedLinearOperators (X)} = x and 1_{RNormedAlgebraOfBoundedLinearOperators (X)} ·x = x and x·(y +z) = x·y +x·z and (y+z)·x = y·x+z·x and a·(x·y) = (a·x)·y and (a·b)·(x·y) = a·x·(b·y) and a·(x+y) = a·x+a·y and (a+b)·x = a·x+b·x and (a·b)·x = a·(b·x) and 1 · x = x.

(20) Let X be a real normed space. Then RNormedAlgebraOfBoundedLinearOperators(X) is real normed space-like, Abelian, add-associative, right zeroed, right complementable, associative, algebra-like, and real linear space-like.

Let us observe that there exists a non empty normed algebra structure which is real normed space-like, Abelian, add-associative, right zeroed, right complementable, associative, algebra-like, real linear space-like, and strict.

A normed algebra is a real normed space-like Abelian add-associative right zeroed right complementable associative algebra-like real linear space-like non empty normed algebra structure.

Let X be a real normed space. Observe that RNormedAlgebraOfBoundedLinearOperators(X) is real nor-

med space-like, Abelian, add-associative, right zeroed, right complementable, associative, algebra-like, and real linear space-like.

Let X be a non empty normed algebra structure. We say that X is Banach Algebra-like1 if and only if:

(Def. 9) For all elements x, y of X holds ∥x · y∥ ∥x∥ · ∥y∥.

We say that X is Banach Algebra-like2 if and only if: (Def. 10) ∥1_{X}∥ = 1.

We say that X is Banach Algebra-like3 if and only if: (Def. 11) For every real number a and for all elements x, y of X holds a · (x · y) =

x · (a · y). Let X be a normed algebra. We say that X is Banach Algebra-like if and

only if the condition (Def. 12) is satisfied. (Def. 12) X is Banach Algebra-like1, Banach Algebra-like2, Banach Algebra-like3,

left unital, left distributive, and complete. Let us mention that every normed algebra which is Banach Algebra-like

is also Banach Algebra-like1, Banach Algebra-like2, Banach Algebra-like3, left distributive, left unital, and complete and every normed algebra which is Banach Algebra-like1, Banach Algebra-like2, Banach Algebra-like3, left distributive, left unital, and complete is also Banach Algebra-like.

108 yasunari shidama

Let X be a non trivial real Banach space. Note that RNormedAlgebraOfBoundedLinearOperators(X) is Banach

Algebra-like.

One can verify that there exists a normed algebra which is Banach Algebralike.

A Banach algebra is a Banach Algebra-like normed algebra.

References

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[2] Grzegorz Bancerek. Sequences of ordinal numbers. Formalized Mathematics, 1(2):281– 290, 1990.

[3] Józef Białas. Group and field definitions. Formalized Mathematics, 1(3):433–439, 1990.

[4] Czesław Byliński. Binary operations. Formalized Mathematics, 1(1):175–180, 1990.

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[6] Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153–164, 1990.

[7] Czesław Byliński. Partial functions. Formalized Mathematics, 1(2):357–367, 1990.

[8] Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47–53, 1990.

[9] Noboru Endou, Yasumasa Suzuki, and Yasunari Shidama. Real linear space of real sequences. Formalized Mathematics, 11(3):249–253, 2003.

[10] Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35–40, 1990.

[11] Jarosław Kotowicz. Monotone real sequences. Subsequences. Formalized Mathematics, 1(3):471–475, 1990.

[12] Jarosław Kotowicz. Real sequences and basic operations on them. Formalized Mathematics, 1(2):269–272, 1990.

[13] Eugeniusz Kusak, Wojciech Leończuk, and Michał Muzalewski. Abelian groups, fields and vector spaces. Formalized Mathematics, 1(2):335–342, 1990.

[14] Henryk Oryszczyszyn and Krzysztof Prażmowski. Real functions spaces. Formalized Mathematics, 1(3):555–561, 1990.

[15] Jan Popiołek. Introduction to Banach and Hilbert spaces - part I. Formalized Mathematics, 2(4):511–516, 1991.

[16] Jan Popiołek. Real normed space. Formalized Mathematics, 2(1):111–115, 1991.

[17] Yasunari Shidama. Banach space of bounded linear operators. Formalized Mathematics, 12(1):39–48, 2003.

[18] Andrzej Trybulec. Subsets of complex numbers. To appear in Formalized Mathematics.

[19] Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics, 1(2):329–334, 1990.

[20] Andrzej Trybulec. Function domains and Frænkel operator. Formalized Mathematics, 1(3):495–500, 1990.

[21] Andrzej Trybulec. Tarski Grothendieck set theory. Formalized Mathematics, 1(1):9–11, 1990.

[22] Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291–296, 1990.

[23] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67–71, 1990.

[24] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73–83, 1990.

[25] Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181–186, 1990.

Complex Linear Space of Complex Sequences

Noboru Endou Gifu National College of Technology

Summary. In this article, we introduce a notion of complex linear space of complex sequence and complex unitary space.

MML Identifier: CSSPACE.

The notation and terminology used here are introduced in the following papers: [18], [21], [22], [17], [5], [6], [10], [3], [7], [16], [9], [12], [19], [4], [1], [11], [15], [14], [2], [20], [13], and [8].

1. Linear Space of Complex Sequence

The non empty set the set of complex sequences is defined by: (Def. 1) For every set x holds x ∈ the set of complex sequences iff x is a complex

sequence. Let z be a set. Let us assume that z ∈ the set of complex sequences. The

functor id_{seq}(z) yields a complex sequence and is defined by: (Def. 2) id_{seq}(z) = z.

Let z be a set. Let us assume that z ∈ ℂ. The functor id_{ℂ}(z) yielding a Complex is defined by:

(Def. 3) id_{ℂ}(z) = z.

One can prove the following propositions:

(1) There exists a binary operation A_{1} on the set of complex sequences such that

(i) for all elements a, b of the set of complex sequences holds A_{1}(a, b) = id_{seq}(a) + id_{seq}(b), and

(ii) A_{1} is commutative and associative.

(2) There exists a function f from [: ℂ, the set of complex sequences :] into the set of complex sequences such that for all sets r, x if r ∈ ℂ and x ∈ the set of complex sequences, then f(⟨⟨r, x⟩⟩) = id_{ℂ}(r) id_{seq}(x).

The binary operation add_{seq} on the set of complex sequences is defined as follows:

(Def. 4) For all elements a, b of the set of complex sequences holds add_{seq}(a,

b) = id_{seq}(a) + id_{seq}(b). The function mult_{seq} from [: ℂ, the set of complex sequences :] into the set of

complex sequences is defined as follows: (Def. 5) For all sets z, x such that z ∈ ℂ and x ∈ the set of complex sequences

holds mult_{seq}(⟨⟨z, x⟩⟩) = id_{ℂ}(z) id_{seq}(x). The element CZeroseq of the set of complex sequences is defined by:

(Def. 6) For every natural number n holds (id_{seq}(CZeroseq))(n) = 0_{ℂ}. One can prove the following propositions:

(3) For every complex sequence x holds id_{seq}(x) = x.

(4) For all vectors v, w of ⟨the set of complex sequences, CZeroseq, add_{seq}, mult_{seq}⟩ holds v + w = id_{seq}(v) + id_{seq}(w).

(5) For every Complex z and for every vector v of ⟨the set of complex sequences, CZeroseq, add_{seq}, mult_{seq}⟩ holds z · v = z id_{seq}(v).

One can check that ⟨the set of complex sequences, CZeroseq, add_{seq}, mult_{seq}⟩ is Abelian.

Next we state several propositions:

(6) For all vectors u, v, w of ⟨the set of complex sequences, CZeroseq, add_{seq}, mult_{seq}⟩ holds (u + v) + w = u + (v + w).

(7) For every vector v of ⟨the set of complex sequences, CZeroseq, add_{seq}, mult_{seq}⟩ holds v + 0_{⟨the set of complex sequences ,CZeroseq ,add_{seq} ,mult_{seq} ⟩}= v.

(8) Let v be a vector of ⟨the set of complex sequences, CZeroseq, add_{seq}, mult_{seq}⟩. Then there exists a vector w of ⟨the set of complex sequences, CZeroseq, add_{seq}, mult_{seq}⟩ such that v + w = 0_{⟨the set of complex sequences ,CZeroseq ,add_{seq} ,mult_{seq} ⟩}.

(9) For every Complex z and for all vectors v, w of ⟨the set of complex sequences, CZeroseq, add_{seq}, mult_{seq}⟩ holds z · (v + w) = z · v + z · w.

(10) For all Complexes z_{1}, z_{2} and for every vector v of ⟨the set of complex sequences, CZeroseq, add_{seq}, mult_{seq}⟩ holds (z_{1} + z_{2}) · v = z_{1} · v + z_{2} · v.

(11) For all Complexes z_{1}, z_{2} and for every vector v of ⟨the set of complex sequences, CZeroseq, add_{seq}, mult_{seq}⟩ holds (z_{1} · z_{2}) · v = z_{1} · (z_{2} · v).

(12) For every vector v of ⟨the set of complex sequences, CZeroseq, add_{seq}, mult_{seq}⟩ holds 1_{ℂ} · v = v.

The complex linear space the linear space of complex sequences is defined as follows:

(Def. 7) The linear space of complex sequences = ⟨the set of complex

sequences, CZeroseq, add_{seq}, mult_{seq}⟩. Let X be a complex linear space and let X_{1} be a subset of X. Let us assume

that X_{1} is linearly closed and non empty. The functor Add (X_{1}, X) yields a binary operation on X_{1} and is defined by:

(Def. 8) Add (X_{1}, X) = (the addition of X)↾[: X_{1}, X_{1} :].

Let X be a complex linear space and let X_{1} be a subset of X. Let us assume that X_{1} is linearly closed and non empty. The functor Mult (X_{1}, X) yields a function from [: ℂ, X_{1} :] into X_{1} and is defined as follows:

(Def. 9) Mult (X_{1}, X) = (the external multiplication of X)↾[: ℂ, X_{1} :].

Let X be a complex linear space and let X_{1} be a subset of X. Let us assume that X_{1} is linearly closed and non empty. The functor Zero (X_{1}, X) yielding an element of X_{1} is defined by:

(Def. 10) Zero (X_{1}, X) = 0_{X}.

One can prove the following proposition

(13) Let V be a complex linear space and V_{1} be a subset of V. Suppose V_{1} is linearly closed and non empty. Then ⟨V_{1}, Zero (V_{1}, V), Add (V_{1}, V), Mult (V_{1}, V)⟩ is a subspace of V.

The subset the set of l2-complex sequences of the linear space of complex sequences is defined by the conditions (Def. 11).

(Def. 11)(i) The set of l2-complex sequences is non empty, and

(ii) for every set x holds x ∈ the set of l2-complex sequences iff x ∈ the set

of complex sequences and | id_{seq}(x)| | id_{seq}(x)| is summable. One can prove the following propositions:

(14) The set of l2-complex sequences is linearly closed and the set of l2complex sequences is non empty.

(15) ⟨the set of l2-complex sequences, Zero (the set of l2-complex sequences, the linear space of complex sequences), Add (the set of l2complex sequences, the linear space of complex sequences), Mult (the set of l2-complex sequences, the linear space of complex sequences)⟩ is a subspace of the linear space of complex sequences.

(16) ⟨the set of l2-complex sequences, Zero (the set of l2-complex sequences, the linear space of complex sequences), Add (the set of l2complex sequences, the linear space of complex sequences), Mult (the set of l2-complex sequences, the linear space of complex sequences)⟩ is a complex linear space.

(17)(i) The carrier of the linear space of complex sequences = the set of

complex sequences, (ii) for every set x holds x is an element of the linear space of complex

sequences iff x is a complex sequence, (iii) for every set x holds x is a vector of the linear space of complex sequ-

ences iff x is a complex sequence, (iv) for every vector u of the linear space of complex sequences holds u =

id_{seq}(u),

(v) for all vectors u, v of the linear space of complex sequences holds u + v = id_{seq}(u) + id_{seq}(v), and

(vi) for every Complex z and for every vector u of the linear space of complex sequences holds z · u = z id_{seq}(u).

2. Unitary Space with Complex Coefficient

We introduce complex unitary space structures which are extensions of CLS structure and are systems

⟨ a carrier, a zero, an addition, an external multiplication, a scalar product ⟩, where the carrier is a set, the zero is an element of the carrier, the addition is a binary operation on the carrier, the external multiplication is a function from [: ℂ, the carrier :] into the carrier, and the scalar product is a function from [: the carrier, the carrier :] into ℂ.

Let us note that there exists a complex unitary space structure which is non empty and strict.

Let D be a non empty set, let Z be an element of D, let a be a binary operation on D, let m be a function from [: ℂ, D:] into D, and let s be a function from [: D, D:] into ℂ. Note that ⟨D, Z, a, m, s⟩ is non empty.

We adopt the following rules: X is a non empty complex unitary space structure, a, b are Complexes, and x, y are points of X.

Let us consider X and let us consider x, y. The functor (x|y) yields a Complex and is defined by:

(Def. 12) (x|y) = (the scalar product of X)(⟨⟨x, y⟩⟩).

Let I_{1} be a non empty complex unitary space structure. We say that I_{1} is complex unitary space-like if and only if the condition (Def. 13) is satisfied.

(Def. 13) Let x, y, w be points of I_{1} and given a. Then (x|x) = 0 iff x = 0_{(I_{1})} and 0 ℜ((x|x)) and 0 = ℑ((x|x)) and (x|y) = \overline{(y|x)} and ((x + y)|w) = (x|w) + (y|w) and ((a · x)|y) = a · (x|y).

Let us note that there exists a non empty complex unitary space structure which is complex unitary space-like, complex linear space-like, Abelian, addassociative, right zeroed, right complementable, and strict.

A complex unitary space is a complex unitary space-like complex linear space-like Abelian add-associative right zeroed right complementable non empty complex unitary space structure.

We use the following convention: X is a complex unitary space and x, y, z, u, v are points of X.

Next we state a number of propositions:

(18) (0_{X}|0_{X}) = 0.

(19) (x|(y + z)) = (x|y) + (x|z).

(20) (x|(a · y)) = \overline{a} · (x|y).

(21) ((a · x)|y) = (x|(\overline{a} · y)).

(22) ((a · x + b · y)|z) = a · (x|z) + b · (y|z).

(23) (x|(a · y + b · z)) = \overline{a} · (x|y) + \overline{b} · (x|z).

(24) ((− x)|y) = (x|− y).

(25) ((− x)|y) = − (x|y).

(26) (x|− y) = − (x|y).

(27) ((− x)|− y) = (x|y).

(28) ((x − y)|z) = (x|z) − (y|z).

(29) (x|(y − z)) = (x|y) − (x|z).

(30) ((x − y)|(u − v)) = ((x|u) − (x|v) − (y|u)) + (y|v).

(31) (0_{X}|x) = 0.

(32) (x|0_{X}) = 0.

(33) ((x + y)|(x + y)) = (x|x) + (x|y) + (y|x) + (y|y).

(34) ((x + y)|(x − y)) = (((x|x) − (x|y)) + (y|x)) − (y|y).

(35) ((x − y)|(x − y)) = ((x|x) − (x|y) − (y|x)) + (y|y).

(36) |(x|x)| = ℜ((x|x)).

(37) |(x|y)| \sqrt{|(x|x)|} · \sqrt{|(y|y)|}.

Let us consider X and let us consider x, y. We say that x, y are orthogonal if and only if:

(Def. 14) (x|y) = 0.

Let us note that the predicate x, y are orthogonal is symmetric.

We now state several propositions:

(38) If x, y are orthogonal, then x, − y are orthogonal.

(39) If x, y are orthogonal, then − x, y are orthogonal.

(40) If x, y are orthogonal, then − x, − y are orthogonal.

(41) x, 0_{X} are orthogonal.

(42) If x, y are orthogonal, then ((x + y)|(x + y)) = (x|x) + (y|y).

(43) If x, y are orthogonal, then ((x − y)|(x − y)) = (x|x) + (y|y).

Let us consider X, x. The functor ∥x∥ yields a real number and is defined as follows:

(Def. 15) ∥x∥ = \sqrt{|(x|x)|}.

We now state several propositions:

(44) ∥x∥ = 0 iff x = 0_{X}.

(45) ∥a · x∥ = |a| · ∥x∥.

(46) 0 ∥x∥.

(47) |(x|y)| ∥x∥ · ∥y∥.

(48) ∥x + y∥ ∥x∥ + ∥y∥.

(49) ∥− x∥ = ∥x∥.

(50) ∥x∥ − ∥y∥ ∥x − y∥.

(51) |∥x∥ − ∥y∥| ∥x − y∥.

Let us consider X, x, y. The functor ρ(x, y) yielding a real number is defined as follows:

(Def. 16) ρ(x, y) = ∥x − y∥.

One can prove the following proposition

(52) ρ(x, y) = ρ(y, x).

Let us consider X, x, y. Let us observe that the functor ρ(x, y) is commutative.

We now state a number of propositions:

(53) ρ(x, x) = 0.

(54) ρ(x, z) ρ(x, y) + ρ(y, z).

(55) x ≠ y iff ρ(x, y) ≠ 0.

(56) ρ(x, y) 0.

(57) x ≠ y iff ρ(x, y) > 0.

(58) ρ(x, y) = \sqrt{|((x − y)|(x − y))|} .

(59) ρ(x + y, u + v) ρ(x, u) + ρ(y, v).

(60) ρ(x − y, u − v) ρ(x, u) + ρ(y, v).

(61) ρ(x − z, y − z) = ρ(x, y).

(62) ρ(x − z, y − z) ρ(z, x) + ρ(z, y). We follow the rules: s_{1}, s_{2}, s_{3}, s_{4} are sequences of X and k, n, m are natural

numbers.

The scheme Ex Seq in CUS deals with a non empty complex unitary space structure A and a unary functor F yielding a point of A, and states that:

There exists a sequence s_{1} of A such that for every n holds s_{1}(n) = F(n)

for all values of the parameters.

Let us consider X and let us consider s_{1}. The functor − s_{1} yielding a sequence of X is defined by:

(Def. 17) For every n holds (− s_{1})(n) = − s_{1}(n).

Let us consider X, let us consider s_{1}, and let us consider x. The functor s_{1} + x yielding a sequence of X is defined by:

(Def. 18) For every n holds (s_{1} + x)(n) = s_{1}(n) + x.

One can prove the following proposition

(63) s_{2} + s_{3} = s_{3} + s_{2}.

Let us consider X, s_{2}, s_{3}. Let us observe that the functor s_{2} + s_{3} is commutative.

One can prove the following propositions:

(64) s_{2} + (s_{3} + s_{4}) = (s_{2} + s_{3}) + s_{4}.

(65) If s_{2} is constant and s_{3} is constant and s_{1} = s_{2} + s_{3}, then s_{1} is constant.

(66) If s_{2} is constant and s_{3} is constant and s_{1} = s_{2} − s_{3}, then s_{1} is constant.

(67) If s_{2} is constant and s_{1} = a · s_{2}, then s_{1} is constant.

(68) s_{1} is constant iff for every n holds s_{1}(n) = s_{1}(n + 1).

(69) s_{1} is constant iff for all n, k holds s_{1}(n) = s_{1}(n + k).

(70) s_{1} is constant iff for all n, m holds s_{1}(n) = s_{1}(m).

(71) s_{2} − s_{3} = s_{2} + − s_{3}.

(72) s_{1} = s_{1} + 0_{X}.

(73) a · (s_{2} + s_{3}) = a · s_{2} + a · s_{3}.

(74) (a + b) · s_{1} = a · s_{1} + b · s_{1}.

(75) (a · b) · s_{1} = a · (b · s_{1}).

(76) 1_{ℂ} · s_{1} = s_{1}.

(77) (− 1_{ℂ}) · s_{1} = − s_{1}.

(78) s_{1} − x = s_{1} + − x.

(79) s_{2} − s_{3} = − (s_{3} − s_{2}) .

(80) s_{1} = s_{1} − 0_{X}.

(81) s_{1} = − − s_{1}.

(82) s_{2} − (s_{3} + s_{4}) = s_{2} − s_{3} − s_{4}.

(83) (s_{2} + s_{3}) − s_{4} = s_{2} + (s_{3} − s_{4}).

(84) s_{2} − (s_{3} − s_{4}) = (s_{2} − s_{3}) + s_{4}.

(85) a · (s_{2} − s_{3}) = a · s_{2} − a · s_{3}.

3. Complex Unitary Space of Complex Sequence

Next we state the proposition

(86) There exists a function f from [: the set of l2-complex sequences, the set of l2-complex sequences :] into ℂ such that for all sets x, y if x ∈ the set of l2-complex sequences and y ∈ the set of l2-complex sequences, then f(⟨⟨x, y⟩⟩) = ∑(id_{seq}(x) \overline{id_{seq}(y)}).

The function scalar_{cl} from [: the set of l2-complex sequences, the set of l2complex sequences :] into ℂ is defined by the condition (Def. 19).

(Def. 19) Let x, y be sets. Suppose x ∈ the set of l2-complex sequences and y ∈ the

set of l2-complex sequences. Then scalar_{cl}(⟨⟨x, y⟩⟩) = ∑(id_{seq}(x) \overline{id_{seq}(y)}). Let us observe that ⟨the set of l2-complex sequences, Zero (the set of l2-

complex sequences, the linear space of complex sequences), Add (the set of l2complex sequences, the linear space of complex sequences), Mult (the set of l2-complex sequences, the linear space of complex sequences), scalar_{cl}⟩ is non empty.

The non empty complex unitary space structure Complexl2-Space is defined by the condition (Def. 20).

(Def. 20) Complexl2-Space = ⟨the set of l2-complex sequences, Zero (the set of l2complex sequences, the linear space of complex sequences), Add (the set of l2-complex sequences, the linear space of complex sequences), Mult (the set of l2-complex sequences, the linear space of complex sequences), scalar_{cl}⟩.

The following propositions are true:

(87) Let l be a complex unitary space structure. Suppose ⟨the carrier of l, the zero of l, the addition of l, the external multiplication of l⟩ is a complex linear space. Then l is a complex linear space.

(88) For every complex sequence s_{1} such that for every natural number n

holds s_{1}(n) = 0_{ℂ} holds s_{1} is summable and ∑ s_{1} = 0_{ℂ}. Let us observe that Complexl2-Space is Abelian, add-associative, right ze-

roed, right complementable, and complex linear space-like.

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complex linear space of complex sequences 117

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[14] Konrad Raczkowski and Andrzej Nędzusiak. Series. Formalized Mathematics, 2(4):449– 452, 1991.

[15] Yasunari Shidama and Artur Korniłowicz. Convergence and the limit of complex sequences. Series. Formalized Mathematics, 6(3):403–410, 1997.

[16] Andrzej Trybulec. Subsets of complex numbers. To appear in Formalized Mathematics.

[17] Andrzej Trybulec. Domains and their Cartesian products. Formalized Mathematics, 1(1):115–122, 1990.

[18] Andrzej Trybulec. Tarski Grothendieck set theory. Formalized Mathematics, 1(1):9–11, 1990.

[19] Andrzej Trybulec and Czesław Byliński. Some properties of real numbers. Formalized Mathematics, 1(3):445–449, 1990.

[20] Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291–296, 1990.

[21] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67–71, 1990.

[22] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73–83, 1990.

Behaviour of an Arc Crossing a Line

Yatsuka Nakamura Shinshu University Nagano

Summary. In two-dimensional Euclidean space, we examine behaviour of an arc when it crosses a vertical line. There are three types when an arc enters into a line, which are: “Left-In”, “Right-In” and “Oscillating-In”. Also, there are three types when an arc goes out from a line, which are: “Left-Out”, “Right-Out” and “Oscillating-Out”. If an arc is a special polygonal arc, there are only two types for each case, entering in and going out. They are “Left-In” and “Right-In” for entering in, and “Left-Out” and “Right-Out” for going out.

MML Identifier: JORDAN20.

The articles [23], [26], [27], [7], [20], [16], [5], [15], [19], [24], [11], [6], [12], [9], [21], [10], [22], [2], [3], [14], [17], [18], [25], [4], [13], [1], and [8] provide the terminology and notation for this paper.

The following propositions are true:

(1) For every subset P of E_{T}^{2} and for all points p_{1}, p_{2}, p of E_{T}^{2} such that P is an arc from p_{1} to p_{2} and p ∈ P holds Segment(P, p_{1}, p_{2}, p, p) = {p}.

(2) For all points p_{1}, p_{2}, p of E_{T}^{2} and for every real number a such that p ∈ L(p_{1}, p_{2}) and (p_{1})_{1} a and (p_{2})_{1} a holds p_{1} a.

(3) For all points p_{1}, p_{2}, p of E_{T}^{2} and for every real number a such that p ∈ L(p_{1}, p_{2}) and (p_{1})_{1} a and (p_{2})_{1} a holds p_{1} a.

(4) For all points p_{1}, p_{2}, p of E_{T}^{2} and for every real number a such that p ∈ L(p_{1}, p_{2}) and (p_{1})_{1} < a and (p_{2})_{1} < a holds p_{1} < a.

(5) For all points p_{1}, p_{2}, p of E_{T}^{2} and for every real number a such that p ∈ L(p_{1}, p_{2}) and (p_{1})_{1} > a and (p_{2})_{1} > a holds p_{1} > a.

In the sequel j is a natural number. Next we state two propositions:

(6) Let f be a S-sequence in ℝ^{2} and p, q be points of E_{T}^{2}. Suppose 1 j and j < lenf and p ∈ L(f, j) and q ∈ L(f, j) and (f_{j})_{2} = (f_{j+1})_{2} and (f_{j})_{1} > (f_{j+1})_{1} and LE p, q, L̃(f), f_{1}, f_{len f}. Then p_{1} q_{1}.

(7) Let f be a S-sequence in ℝ^{2} and p, q be points of E_{T}^{2}. Suppose 1 j and j < lenf and p ∈ L(f, j) and q ∈ L(f, j) and (f_{j})_{2} = (f_{j+1})_{2} and (f_{j})_{1} < (f_{j+1})_{1} and LE p, q, L̃(f), f_{1}, f_{len f}. Then p_{1} q_{1}.

Let P be a subset of E_{T}^{2}, let p_{1}, p_{2}, p be points of E_{T}^{2}, and let e be a real number. We say that p is LIn of P, p_{1}, p_{2}, e if and only if the conditions (Def. 1) are satisfied.

(Def. 1)(i) P is an arc from p_{1} to p_{2}, (ii) p ∈ P, (iii) p_{1} = e, and (iv) there exists a point p_{4} of E_{T}^{2} such that (p_{4})_{1} < e and LE p_{4}, p, P, p_{1},

p_{2} and for every point p_{5} of E_{T}^{2} such that LE p_{4}, p_{5}, P, p_{1}, p_{2} and LE p_{5}, p, P, p_{1}, p_{2} holds (p_{5})_{1} e.

We say that p is RIn of P, p_{1}, p_{2}, e if and only if the conditions (Def. 2) are satisfied.

(Def. 2)(i) P is an arc from p_{1} to p_{2}, (ii) p ∈ P, (iii) p_{1} = e, and (iv) there exists a point p_{4} of E_{T}^{2} such that (p_{4})_{1} > e and LE p_{4}, p, P, p_{1},

p_{2} and for every point p_{5} of E_{T}^{2} such that LE p_{4}, p_{5}, P, p_{1}, p_{2} and LE p_{5}, p, P, p_{1}, p_{2} holds (p_{5})_{1} e.

We say that p is LOut of P, p_{1}, p_{2}, e if and only if the conditions (Def. 3) are satisfied.

(Def. 3)(i) P is an arc from p_{1} to p_{2}, (ii) p ∈ P, (iii) p_{1} = e, and (iv) there exists a point p_{4} of E_{T}^{2} such that (p_{4})_{1} < e and LE p, p_{4}, P, p_{1},

p_{2} and for every point p_{5} of E_{T}^{2} such that LE p_{5}, p_{4}, P, p_{1}, p_{2} and LE p, p_{5}, P, p_{1}, p_{2} holds (p_{5})_{1} e.

We say that p is ROut of P, p_{1}, p_{2}, e if and only if the conditions (Def. 4) are satisfied.

(Def. 4)(i) P is an arc from p_{1} to p_{2}, (ii) p ∈ P, (iii) p_{1} = e, and (iv) there exists a point p_{4} of E_{T}^{2} such that (p_{4})_{1} > e and LE p, p_{4}, P, p_{1},

p_{2} and for every point p_{5} of E_{T}^{2} such that LE p_{5}, p_{4}, P, p_{1}, p_{2} and LE p, p_{5}, P, p_{1}, p_{2} holds (p_{5})_{1} e.

We say that p is OsIn of P, p_{1}, p_{2}, e if and only if the conditions (Def. 5) are satisfied.

(Def. 5)(i) P is an arc from p_{1} to p_{2}, (ii) p ∈ P, (iii) p_{1} = e, and (iv) there exists a point p_{7} of E_{T}^{2} such that LE p_{7}, p, P, p_{1}, p_{2} and for every

point p_{8} of E_{T}^{2} such that LE p_{7}, p_{8}, P, p_{1}, p_{2} and LE p_{8}, p, P, p_{1}, p_{2} holds (p_{8})_{1} = e and for every point p_{4} of E_{T}^{2} such that LE p_{4}, p_{7}, P, p_{1}, p_{2} and p_{4} ≠ p_{7} holds there exists a point p_{5} of E_{T}^{2} such that LE p_{4}, p_{5}, P, p_{1}, p_{2} and LE p_{5}, p_{7}, P, p_{1}, p_{2} and (p_{5})_{1} > e and there exists a point p_{6} of E_{T}^{2} such that LE p_{4}, p_{6}, P, p_{1}, p_{2} and LE p_{6}, p_{7}, P, p_{1}, p_{2} and (p_{6})_{1} < e.

We say that p is OsOut of P, p_{1}, p_{2}, e if and only if the conditions (Def. 6) are satisfied.

(Def. 6)(i) P is an arc from p_{1} to p_{2}, (ii) p ∈ P, (iii) p_{1} = e, and (iv) there exists a point p_{7} of E_{T}^{2} such that LE p, p_{7}, P, p_{1}, p_{2} and for every

point p_{8} of E_{T}^{2} such that LE p_{8}, p_{7}, P, p_{1}, p_{2} and LE p, p_{8}, P, p_{1}, p_{2} holds (p_{8})_{1} = e and for every point p_{4} of E_{T}^{2} such that LE p_{7}, p_{4}, P, p_{1}, p_{2} and p_{4} ≠ p_{7} holds there exists a point p_{5} of E_{T}^{2} such that LE p_{5}, p_{4}, P, p_{1}, p_{2} and LE p_{7}, p_{5}, P, p_{1}, p_{2} and (p_{5})_{1} > e and there exists a point p_{6} of E_{T}^{2} such that LE p_{6}, p_{4}, P, p_{1}, p_{2} and LE p_{7}, p_{6}, P, p_{1}, p_{2} and (p_{6})_{1} < e.

We now state a number of propositions:

(8) Let P be a subset of E_{T}^{2}, p_{1}, p_{2}, p be points of E_{T}^{2}, and e be a real number. Suppose P is an arc from p_{1} to p_{2} and (p_{1})_{1} e and (p_{2})_{1} e. Then there exists a point p_{3} of E_{T}^{2} such that p_{3} ∈ P and (p_{3})_{1} = e.

(9) Let P be a non empty subset of E_{T}^{2}, p_{1}, p_{2}, p be points of E_{T}^{2}, and e be a real number. Suppose P is an arc from p_{1} to p_{2} and (p_{1})_{1} < e and (p_{2})_{1} > e and p ∈ P and p_{1} = e. Then p is LIn of P, p_{1}, p_{2}, e, RIn of P, p_{1}, p_{2}, e, and OsIn of P, p_{1}, p_{2}, e.

(10) Let P be a non empty subset of E_{T}^{2}, p_{1}, p_{2}, p be points of E_{T}^{2}, and e be a real number. Suppose P is an arc from p_{1} to p_{2} and (p_{1})_{1} < e and (p_{2})_{1} > e and p ∈ P and p_{1} = e. Then p is LOut of P, p_{1}, p_{2}, e, ROut of P, p_{1}, p_{2}, e, and OsOut of P, p_{1}, p_{2}, e.

(11) For every subset P of 𝕀 and for every real number s such that P = [0, s[ holds P is open.

(12) For every subset P of 𝕀 and for every real number s such that P = ]s, 1] holds P is open.

(13) Let P be a non empty subset of E_{T}^{2}, P_{1} be a subset of (E_{T}^{2})↾P, Q be a subset of 𝕀, f be a map from 𝕀 into (E_{T}^{2})↾P, and s be a real number. Suppose s 1 and P_{1} = {q_{0}; q_{0} ranges over points of E_{T}^{2}: ⋁_{s_{1} : real number} (0 s_{1} ∧ s_{1} < s ∧ q_{0} = f(s_{1}))} and Q = [0, s[. Then f^{∘}Q = P_{1}.

(14) Let P be a non empty subset of E_{T}^{2}, P_{1} be a subset of (E_{T}^{2})↾P, Q be a subset of 𝕀, f be a map from 𝕀 into (E_{T}^{2})↾P, and s be a real number. Suppose s 0 and P_{1} = {q_{0}; q_{0} ranges over points of E_{T}^{2}: ⋁_{s_{1} : real number} (s < s_{1} ∧ s_{1} 1 ∧ q_{0} = f(s_{1}))} and Q = ]s, 1]. Then f^{∘}Q = P_{1}.

(15) Let P be a non empty subset of E_{T}^{2}, P_{1} be a subset of (E_{T}^{2})↾P, f be a map from 𝕀 into (E_{T}^{2})↾P, and s be a real number. Suppose s 1 and f is a homeomorphism and P_{1} = {q_{0}; q_{0} ranges over points of E_{T}^{2}: ⋁_{s_{1} : real number} (0 s_{1} ∧ s_{1} < s ∧ q_{0} = f(s_{1}))}. Then P_{1} is open.

(16) Let P be a non empty subset of E_{T}^{2}, P_{1} be a subset of (E_{T}^{2})↾P, f be a map from 𝕀 into (E_{T}^{2})↾P, and s be a real number. Suppose s 0 and f is a homeomorphism and P_{1} = {q_{0}; q_{0} ranges over points of E_{T}^{2}: ⋁_{s_{1} : real number} (s < s_{1} ∧ s_{1} 1 ∧ q_{0} = f(s_{1}))}. Then P_{1} is open.

(17) Let T be a non empty topological structure, Q_{1}, Q_{2} be subsets of T, and p_{1}, p_{2} be points of T. Suppose Q_{1} ∩ Q_{2} = ∅ and Q_{1} ∪ Q_{2} = the carrier of T and p_{1} ∈ Q_{1} and p_{2} ∈ Q_{2} and Q_{1} is open and Q_{2} is open. Then it is not true that there exists a map P from 𝕀 into T such that P is continuous and P (0) = p_{1} and P (1) = p_{2}.

(18) Let P be a non empty subset of E_{T}^{2}, Q be a subset of (E_{T}^{2})↾P, and p_{1}, p_{2}, q be points of E_{T}^{2}. Suppose P is an arc from p_{1} to p_{2} and q ∈ P and q ≠ p_{1} and q ≠ p_{2} and Q = P \ {q}. Then Q is not connected and it is not true that there exists a map R from 𝕀 into (E_{T}^{2})↾P ↾Q such that R is continuous and R(0) = p_{1} and R(1) = p_{2}.

(19) Let P be a non empty subset of E_{T}^{2} and p_{1}, p_{2}, q_{1}, q_{2} be points of E_{T}^{2}. Suppose P is an arc from p_{1} to p_{2} and q_{1} ∈ P and q_{2} ∈ P. Then LE q_{1}, q_{2}, P, p_{1}, p_{2} or LE q_{2}, q_{1}, P, p_{1}, p_{2}.

(20) Let P be a non empty subset of E_{T}^{2} and p_{1}, p_{2}, q_{1} be points of E_{T}^{2}. Suppose P is an arc from p_{1} to p_{2} and q_{1} ∈ P and p_{1} ≠ q_{1}. Then Segment(P, p_{1}, p_{2}, p_{1}, q_{1}) is an arc from p_{1} to q_{1}.

(21) Let n be a natural number, p_{1}, p_{2} be points of E_{T}^{n}, and P, P_{1} be non empty subsets of E_{T}^{n}. If P is an arc from p_{1} to p_{2} and P_{1} is an arc from p_{1} to p_{2} and P_{1} ⊆ P, then P_{1} = P.

(22) Let P be a non empty subset of E_{T}^{2} and p_{1}, p_{2}, q_{1} be points of E_{T}^{2}. Suppose P is an arc from p_{1} to p_{2} and q_{1} ∈ P and p_{2} ≠ q_{1}. Then Segment(P, p_{1}, p_{2}, q_{1}, p_{2}) is an arc from q_{1} to p_{2}.

(23) Let P be a non empty subset of E_{T}^{2} and p_{1}, p_{2}, q_{1}, q_{2}, q_{3} be points of E_{T}^{2}. Suppose P is an arc from p_{1} to p_{2} and LE q_{1}, q_{2}, P, p_{1}, p_{2} and LE q_{2}, q_{3}, P, p_{1}, p_{2}. Then Segment(P, p_{1}, p_{2}, q_{1}, q_{2}) ∪ Segment(P, p_{1}, p_{2}, q_{2}, q_{3}) = Segment(P, p_{1}, p_{2}, q_{1}, q_{3}).

(24) Let P be a non empty subset of E_{T}^{2} and p_{1}, p_{2}, q_{1}, q_{2}, q_{3} be points of E_{T}^{2}. Suppose P is an arc from p_{1} to p_{2} and LE q_{1}, q_{2}, P, p_{1}, p_{2} and LE q_{2}, q_{3}, P, p_{1}, p_{2}. Then Segment(P, p_{1}, p_{2}, q_{1}, q_{2}) ∩ Segment(P, p_{1}, p_{2}, q_{2}, q_{3}) = {q_{2}}.

(25) For every non empty subset P of E_{T}^{2} and for all points p_{1}, p_{2} of E_{T}^{2} such that P is an arc from p_{1} to p_{2} holds Segment(P, p_{1}, p_{2}, p_{1}, p_{2}) = P.

(26) Let T be a non empty topological space, w_{1}, w_{2}, w_{3} be points of T, and h_{1}, h_{2} be maps from 𝕀 into T. Suppose h_{1} is continuous and w_{1} = h_{1}(0) and w_{2} = h_{1}(1) and h_{2} is continuous and w_{2} = h_{2}(0) and w_{3} = h_{2}(1). Then there exists a map h_{3} from 𝕀 into T such that h_{3} is continuous and w_{1} = h_{3}(0) and w_{3} = h_{3}(1).

(27) Let T be a non empty topological space, a, b, c be points of T, G_{1} be a path from a to b, and G_{2} be a path from b to c. Suppose G_{1} is continuous and G_{2} is continuous and G_{1}(0) = a and G_{1}(1) = b and G_{2}(0) = b and G_{2}(1) = c. Then G_{1} + G_{2} is continuous and (G_{1} + G_{2})(0) = a and (G_{1} + G_{2})(1) = c.

(28) Let P, Q_{1} be non empty subsets of E_{T}^{2} and p_{1}, p_{2}, q_{1}, q_{2} be points of E_{T}^{2}. Suppose P is an arc from p_{1} to p_{2} and Q_{1} is an arc from q_{1} to q_{2} and LE q_{1}, q_{2}, P, p_{1}, p_{2} and Q_{1} ⊆ P. Then Q_{1} = Segment(P, p_{1}, p_{2}, q_{1}, q_{2}).

(29) Let P be a non empty subset of E_{T}^{2}, p_{1}, p_{2}, q_{1}, q_{2}, p be points of E_{T}^{2}, and e be a real number. Suppose (p_{1})_{1} < e and (p_{2})_{1} > e and q_{1} is LIn of P, p_{1}, p_{2}, e and (q_{2})_{1} = e and L(q_{1}, q_{2}) ⊆ P and p ∈ L(q_{1}, q_{2}). Then p is LIn of P, p_{1}, p_{2}, e.

(30) Let P be a non empty subset of E_{T}^{2}, p_{1}, p_{2}, q_{1}, q_{2}, p be points of E_{T}^{2}, and e be a real number. Suppose (p_{1})_{1} < e and (p_{2})_{1} > e and q_{1} is RIn of P, p_{1}, p_{2}, e and (q_{2})_{1} = e and L(q_{1}, q_{2}) ⊆ P and p ∈ L(q_{1}, q_{2}). Then p is RIn of P, p_{1}, p_{2}, e.

(31) Let P be a non empty subset of E_{T}^{2}, p_{1}, p_{2}, q_{1}, q_{2}, p be points of E_{T}^{2}, and e be a real number. Suppose (p_{1})_{1} < e and (p_{2})_{1} > e and q_{1} is LOut of P, p_{1}, p_{2}, e and (q_{2})_{1} = e and L(q_{1}, q_{2}) ⊆ P and p ∈ L(q_{1}, q_{2}). Then p is LOut of P, p_{1}, p_{2}, e.

(32) Let P be a non empty subset of E_{T}^{2}, p_{1}, p_{2}, q_{1}, q_{2}, p be points of E_{T}^{2}, and e be a real number. Suppose (p_{1})_{1} < e and (p_{2})_{1} > e and q_{1} is ROut of P, p_{1}, p_{2}, e and (q_{2})_{1} = e and L(q_{1}, q_{2}) ⊆ P and p ∈ L(q_{1}, q_{2}). Then p is ROut of P, p_{1}, p_{2}, e.

(33) Let P be a non empty subset of E_{T}^{2}, p_{1}, p_{2}, p be points of E_{T}^{2}, and e be a real number. Suppose P is a special polygonal arc joining p_{1} and p_{2} and (p_{1})_{1} < e and (p_{2})_{1} > e and p ∈ P and p_{1} = e. Then p is LIn of P, p_{1}, p_{2}, e and RIn of P, p_{1}, p_{2}, e.

(34) Let P be a non empty subset of E_{T}^{2}, p_{1}, p_{2}, p be points of E_{T}^{2}, and e be a real number. Suppose P is a special polygonal arc joining p_{1} and p_{2} and 124 yatsuka nakamura

(p_{1})_{1} < e and (p_{2})_{1} > e and p ∈ P and p_{1} = e. Then p is LOut of P, p_{1}, p_{2}, e and ROut of P, p_{1}, p_{2}, e.

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Some Set Series in Finite Topological Spaces. Fundamental Concepts for Image Processing

Masami Tanaka Shinshu University Nagano Yatsuka Nakamura Shinshu University Nagano

Summary. First we give a definition of “inflation” of a set in finite topological spaces. Then a concept of “deflation” of a set is also defined. In the remaining part, we give a concept of the “set series” for a subset of a finite topological space. Using this, we can define a series of neighbourhoods for each point in the space. The work is done according to [7].

MML Identifier: FINTOPO3.

The articles [9], [5], [10], [2], [8], [1], [12], [11], [3], [4], and [6] provide the notation and terminology for this paper.

We adopt the following rules: T denotes a non empty finite topology space, A, B denote subsets of T, and x, y denote elements of T.

Let us consider T and let A be a subset of T. The functor A^{d} yields a subset of T and is defined by:

(Def. 1) A^{d} = {x; x ranges over elements of T: ⋀_{y: element of T} (y ∈ A^{c} ⇒ x ∉

U(y))}. We now state a number of propositions:

(1) If T is filled, then A ⊆ A^{f}.

(2) x ∈ A^{d} iff for every y such that y ∈ A^{c} holds x ∉ U(y).

(3) If T is filled, then A^{d} ⊆ A.

(4) A^{d} = ((A^{c})^{f})^{c}.

(5) If A ⊆ B, then A^{f} ⊆ B^{f}.

(6) If A ⊆ B, then A^{d} ⊆ B^{d}.

(7) (A ∩ B)^{b} ⊆ A^{b} ∩ B^{b}.

(8) (A ∪ B)^{b} = A^{b} ∪ B^{b}.

(9) A^{i} ∪ B^{i} ⊆ (A ∪ B)^{i}.

(10) A^{i} ∩ B^{i} = (A ∩ B)^{i}.

(11) A^{f} ∪ B^{f} = (A ∪ B)^{f}.

(12) A^{d} ∩ B^{d} = A ∩ B^{d}.

Let T be a non empty finite topology space and let A be a subset of T. The functor Fcl(A) yields a function from ℕ into 2^{the carrier of T} and is defined as follows:

(Def. 2) For every natural number n and for every subset B of T such that

B = (Fcl(A))(n) holds (Fcl(A))(n + 1) = B^{b} and (Fcl(A))(0) = A. Let T be a non empty finite topology space, let A be a subset of T, and let n

be a natural number. The functor Fcl(A, n) yields a subset of T and is defined by:

(Def. 3) Fcl(A, n) = (Fcl(A))(n).

Let T be a non empty finite topology space and let A be a subset of T. The functor Fint(A) yields a function from ℕ into 2^{the carrier of T} and is defined by:

(Def. 4) For every natural number n and for every subset B of T such that

B = (Fint(A))(n) holds (Fint(A))(n + 1) = B^{i} and (Fint(A))(0) = A. Let T be a non empty finite topology space, let A be a subset of T, and let n

be a natural number. The functor Fint(A, n) yields a subset of T and is defined as follows:

(Def. 5) Fint(A, n) = (Fint(A))(n).

The following propositions are true:

(13) For every natural number n holds Fcl(A, n + 1) = (Fcl(A, n))^{b}.

(14) Fcl(A, 0) = A.

(15) Fcl(A, 1) = A^{b}.

(16) Fcl(A, 2) = (A^{b})^{b}.

(17) For every natural number n holds Fcl(A ∪ B, n) = Fcl(A, n) ∪ Fcl(B, n).

(18) For every natural number n holds Fint(A, n + 1) = (Fint(A, n))^{i}.

(19) Fint(A, 0) = A.

(20) Fint(A, 1) = A^{i}.

(21) Fint(A, 2) = (A^{i})^{i}.

(22) For every natural number n holds Fint(A ∩ B, n) = Fint(A, n) ∩ Fint(B, n).

(23) If T is filled, then for every natural number n holds A ⊆ Fcl(A, n).

(24) If T is filled, then for every natural number n holds Fint(A, n) ⊆ A.

(25) If T is filled, then for every natural number n holds Fcl(A, n) ⊆ Fcl(A, n + 1).

(26) If T is filled, then for every natural number n holds Fint(A, n + 1) ⊆ Fint(A, n).

(27) For every natural number n holds (Fint(A^{c}, n))^{c} = Fcl(A, n).

(28) For every natural number n holds (Fcl(A^{c}, n))^{c} = Fint(A, n).

(29) For every natural number n holds Fcl(A, n) ∪ Fcl(B, n) = (Fint((A ∪ B)^{c}, n))^{c}.

(30) For every natural number n holds Fint(A, n) ∩ Fint(B, n) = (Fcl((A ∩

B)^{c}, n))^{c}. Let T be a non empty finite topology space and let A be a subset of T. The

functor Finf(A) yielding a function from ℕ into 2^{the carrier of T} is defined by: (Def. 6) For every natural number n and for every subset B of T such that

B = (Finf(A))(n) holds (Finf(A))(n + 1) = B^{f} and (Finf(A))(0) = A. Let T be a non empty finite topology space, let A be a subset of T, and let

n be a natural number. The functor Finf(A, n) yielding a subset of T is defined as follows:

(Def. 7) Finf(A, n) = (Finf(A))(n).

Let T be a non empty finite topology space and let A be a subset of T. The functor Fdfl(A) yields a function from ℕ into 2^{the carrier of T} and is defined as follows:

(Def. 8) For every natural number n and for every subset B of T such that

B = (Fdfl(A))(n) holds (Fdfl(A))(n + 1) = B^{d} and (Fdfl(A))(0) = A. Let T be a non empty finite topology space, let A be a subset of T, and let n

be a natural number. The functor Fdfl(A, n) yields a subset of T and is defined as follows:

(Def. 9) Fdfl(A, n) = (Fdfl(A))(n).

Next we state a number of propositions:

(31) For every natural number n holds Finf(A, n + 1) = (Finf(A, n))^{f}.

(32) Finf(A, 0) = A.

(33) Finf(A, 1) = A^{f}.

(34) Finf(A, 2) = (A^{f})^{f}.

(35) For every natural number n holds Finf(A ∪ B, n) = Finf(A, n) ∪ Finf(B, n).

(36) If T is filled, then for every natural number n holds A ⊆ Finf(A, n).

(37) If T is filled, then for every natural number n holds Finf(A, n) ⊆ Finf(A, n + 1).

(38) For every natural number n holds Fdfl(A, n + 1) = Fdfl(A, n)^{d}.

(39) Fdfl(A, 0) = A.

(40) Fdfl(A, 1) = A^{d}.

(41) Fdfl(A, 2) = (A^{d})^{d}.

(42) For every natural number n holds Fdfl(A ∩ B, n) = Fdfl(A, n) ∩ Fdfl(B, n).

(43) If T is filled, then for every natural number n holds Fdfl(A, n) ⊆ A.

(44) If T is filled, then for every natural number n holds Fdfl(A, n + 1) ⊆ Fdfl(A, n).

(45) For every natural number n holds Fdfl(A, n) = (Finf(A^{c}, n))^{c}.

(46) For every natural number n holds Fdfl(A, n) ∩ Fdfl(B, n) = (Finf((A ∩

B)^{c}, n))^{c}. Let T be a non empty finite topology space, let n be a natural number, and

let x be an element of T. The functor U(x, n) yields a subset of T and is defined as follows:

(Def. 10) U(x, n) = Finf(U(x), n).

Next we state two propositions:

(47) U(x, 0) = U(x).

(48) For every natural number n holds U(x, n + 1) = (U(x, n))^{f}. Let S, T be non empty finite topology spaces. We say that S, T are mutually

symmetric if and only if the conditions (Def. 11) are satisfied. (Def. 11)(i) The carrier of S = the carrier of T, and

(ii) for all sets x, y such that x ∈ the carrier of S and y ∈ the carrier of T holds y ∈ (the neighbour-map of S)(x) iff x ∈ (the neighbour-map of T)(y).

Let us note that the predicate S, T are mutually symmetric is symmetric.

References

[1] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41–46, 1990.

[2] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91–96, 1990.

[3] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55– 65, 1990.

[4] Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153–164, 1990.

[5] Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47–53, 1990.

[6] Hiroshi Imura and Masayoshi Eguchi. Finite topological spaces. Formalized Mathematics, 3(2):189–193, 1992.

[7] Yatsuka Nakamura. Finite topology concept for discrete spaces. In H. Umegaki, editor, Proceedings of the Eleventh Symposium on Applied Functional Analysis, pages 111–116, Noda-City, Chiba, Japan, 1988. Science University of Tokyo.

[8] Andrzej Trybulec. Subsets of complex numbers. To appear in Formalized Mathematics.

[9] Andrzej Trybulec. Tarski Grothendieck set theory. Formalized Mathematics, 1(1):9–11, 1990.

[10] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67–71, 1990.

some set series … 129

[11] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73–83, 1990.

[12] Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181–186, 1990.

The Series on Banach Algebra

Yasunari Shidama Shinshu University Nagano

Summary. In this article, the basic properties of the series on Banach algebra are described. The Neumann series is introduced in the last section.

MML Identifier: LOPBAN 3.

The notation and terminology used in this paper are introduced in the following articles: [19], [21], [22], [4], [5], [3], [2], [18], [6], [1], [20], [10], [11], [12], [17], [9], [7], [8], [14], [13], [15], and [16].

1. Basic Properties of Sequences of Norm Space

Let X be a non empty normed structure and let s_{1} be a sequence of X. The functor (∑_{α=0}^{κ}(s_{1})(α))_{κ∈ℕ} yielding a sequence of X is defined as follows:

(Def. 1) (∑_{α=0}^{κ}(s_{1})(α))_{κ∈ℕ}(0) = s_{1}(0) and for every natural number n holds

(∑_{α=0}^{κ}(s_{1})(α))_{κ∈ℕ}(n + 1) = (∑_{α=0}^{κ}(s_{1})(α))_{κ∈ℕ}(n) + s_{1}(n + 1). One can prove the following proposition

(1) Let X be an add-associative right zeroed right complementable non empty normed structure and s_{1} be a sequence of X. Suppose that for every natural number n holds s_{1}(n) = 0_{X}. Let m be a natural number. Then (∑_{α=0}^{κ}(s_{1})(α))_{κ∈ℕ}(m) = 0_{X}.

Let X be a real normed space and let s_{1} be a sequence of X. We say that s_{1} is summable if and only if:

(Def. 2) (∑_{α=0}^{κ}(s_{1})(α))_{κ∈ℕ} is convergent.

Let X be a real normed space. One can verify that there exists a sequence of X which is summable.

Let X be a real normed space and let s_{1} be a sequence of X. The functor ∑ s_{1} yields an element of X and is defined by:

(Def. 3) ∑ s_{1} = lim((∑_{α=0}^{κ}(s_{1})(α))_{κ∈ℕ}).

Let X be a real normed space and let s_{1} be a sequence of X. We say that s_{1} is norm-summable if and only if:

(Def. 4) ∥s_{1}∥ is summable.

Next we state several propositions:

(2) For every real normed space X and for every sequence s_{1} of X and for every natural number m holds 0 ∥s_{1}∥(m).

(3) For every real normed space X and for all elements x, y, z of X holds ∥x − y∥ = ∥(x − z) + (z − y)∥.

(4) Let X be a real normed space and s_{1} be a sequence of X. Suppose s_{1} is convergent. Let s be a real number. Suppose 0 < s. Then there exists a natural number n such that for every natural number m if n m, then ∥s_{1}(m) − s_{1}(n)∥ < s.

(5) Let X be a real normed space and s_{1} be a sequence of X. Then s_{1} is Cauchy sequence by norm if and only if for every real number p such that p > 0 there exists a natural number n such that for every natural number m such that n m holds ∥s_{1}(m) − s_{1}(n)∥ < p.

(6) Let X be a real normed space and s_{1} be a sequence of X. Suppose that for every natural number n holds s_{1}(n) = 0_{X}. Let m be a natural number. Then (∑_{α=0}^{κ}∥s_{1}∥(α))_{κ∈ℕ}(m) = 0.

Let X be a real normed space and let s_{1} be a sequence of X. Let us observe that s_{1} is constant if and only if:

(Def. 5) There exists an element r of X such that for every natural number n

holds s_{1}(n) = r. Let X be a real normed space, let s_{1} be a sequence of X, and let k be a

natural number. The functor s_{1}↑k yielding a sequence of X is defined as follows: (Def. 6) For every natural number n holds (s_{1} ↑ k)(n) = s_{1}(n + k).

Let X be a non empty 1-sorted structure, let N_{1} be an increasing sequence of naturals, and let s_{1} be a sequence of X. Then s_{1} · N_{1} is a function from ℕ into the carrier of X.

Let X be a non empty 1-sorted structure, let N_{1} be an increasing sequence of naturals, and let s_{1} be a sequence of X. Then s_{1} · N_{1} is a sequence of X.

Let X be a real normed space and let s_{1}, s_{2} be sequences of X. We say that s_{1} is a subsequence of s_{2} if and only if:

(Def. 7) There exists an increasing sequence N_{1} of naturals such that s_{1} = s_{2}·N_{1}. Next we state a number of propositions:

(7) Let X be a non empty 1-sorted structure, s_{1} be a sequence of X, N_{1} be an increasing sequence of naturals, and n be a natural number. Then (s_{1} · N_{1})(n) = s_{1}(N_{1}(n)).

(8) For every real normed space X and for every sequence s_{1} of X holds s_{1} ↑ 0 = s_{1}.

(9) For every real normed space X and for every sequence s_{1} of X and for all natural numbers k, m holds s_{1} ↑ k ↑ m = s_{1} ↑ m ↑ k.

(10) For every real normed space X and for every sequence s_{1} of X and for all natural numbers k, m holds s_{1} ↑ k ↑ m = s_{1} ↑ (k + m).

(11) Let X be a real normed space and s_{1}, s_{2} be sequences of X. If s_{2} is a subsequence of s_{1} and s_{1} is convergent, then s_{2} is convergent.

(12) Let X be a real normed space and s_{1}, s_{2} be sequences of X. If s_{2} is a subsequence of s_{1} and s_{1} is convergent, then lims_{2} = lims_{1}.

(13) Let X be a real normed space, s_{1} be a sequence of X, and k be a natural number. Then s_{1} ↑ k is a subsequence of s_{1}.

(14) Let X be a real normed space, s_{1}, s_{2} be sequences of X, and k be a natural number. If s_{1} is convergent, then s_{1} ↑k is convergent and lim(s_{1} ↑ k) = lims_{1}.

(15) Let X be a real normed space and s_{1}, s_{2} be sequences of X. Suppose s_{1} is convergent and there exists a natural number k such that s_{1} = s_{2} ↑ k. Then s_{2} is convergent.

(16) Let X be a real normed space and s_{1}, s_{2} be sequences of X. Suppose s_{1} is convergent and there exists a natural number k such that s_{1} = s_{2} ↑ k. Then lims_{2} = lims_{1}.

(17) For every real normed space X and for every sequence s_{1} of X such that s_{1} is constant holds s_{1} is convergent.

(18) Let X be a real normed space and s_{1} be a sequence of X. If for every

natural number n holds s_{1}(n) = 0_{X}, then s_{1} is norm-summable. Let X be a real normed space. Observe that there exists a sequence of X

which is norm-summable.

Next we state three propositions:

(19) Let X be a real normed space and s be a sequence of X. If s is summable, then s is convergent and lims = 0_{X}.

(20) For every real normed space X and for all sequences s_{3}, s_{4} of X holds (∑_{α=0}^{κ}(s_{3})(α))_{κ∈ℕ} + (∑_{α=0}^{κ}(s_{4})(α))_{κ∈ℕ} = (∑_{α=0}^{κ}(s_{3} + s_{4})(α))_{κ∈ℕ}.

(21) For every real normed space X and for all sequences s_{3}, s_{4} of X holds

(∑_{α=0}^{κ}(s_{3})(α))_{κ∈ℕ} − (∑_{α=0}^{κ}(s_{4})(α))_{κ∈ℕ} = (∑_{α=0}^{κ}(s_{3} − s_{4})(α))_{κ∈ℕ}. Let X be a real normed space and let s_{1} be a norm-summable sequence of

X. Observe that ∥s_{1}∥ is summable.

Let X be a real normed space. One can check that every sequence of X which is summable is also convergent.

The following propositions are true:

(22) Let X be a real normed space and s_{2}, s_{5} be sequences of X. If s_{2} is summable and s_{5} is summable, then s_{2} +s_{5} is summable and ∑(s_{2} +s_{5}) = ∑ s_{2} + ∑ s_{5}.

(23) Let X be a real normed space and s_{2}, s_{5} be sequences of X. If s_{2} is summable and s_{5} is summable, then s_{2} −s_{5} is summable and ∑(s_{2} −s_{5}) = ∑ s_{2} − ∑ s_{5}.

Let X be a real normed space and let s_{2}, s_{5} be summable sequences of X. One can verify that s_{2} + s_{5} is summable and s_{2} − s_{5} is summable.

We now state two propositions:

(24) For every real normed space X and for every sequence s_{1} of X and for every real number z holds (∑_{α=0}^{κ}(z · s_{1})(α))_{κ∈ℕ} = z · (∑_{α=0}^{κ}(s_{1})(α))_{κ∈ℕ}.

(25) Let X be a real normed space, s_{1} be a summable sequence of X, and z

be a real number. Then z · s_{1} is summable and ∑(z · s_{1}) = z · ∑ s_{1}. Let X be a real normed space, let z be a real number, and let s_{1} be a

summable sequence of X. Observe that z · s_{1} is summable.

One can prove the following two propositions:

(26) Let X be a real normed space and s, s_{3} be sequences of X. If for every natural number n holds s_{3}(n) = s(0), then (∑_{α=0}^{κ}(s ↑ 1)(α))_{κ∈ℕ} = (∑_{α=0}^{κ} s(α))_{κ∈ℕ} ↑ 1 − s_{3}.

(27) Let X be a real normed space and s be a sequence of X. If s is summable,

then for every natural number n holds s ↑ n is summable. Let X be a real normed space, let s_{1} be a summable sequence of X, and let

n be a natural number. Observe that s_{1} ↑ n is summable.

Next we state the proposition

(28) Let X be a real normed space and s_{1} be a sequence of X. Then

(∑_{α=0}^{κ}∥s_{1}∥(α))_{κ∈ℕ} is upper bounded if and only if s_{1} is norm-summable. Let X be a real normed space and let s_{1} be a norm-summable sequence of

X. One can check that (∑_{α=0}^{κ}∥s_{1}∥(α))_{κ∈ℕ} is upper bounded.

One can prove the following propositions:

(29) Let X be a real Banach space and s_{1} be a sequence of X. Then s_{1} is summable if and only if for every real number p such that 0 < p there exists a natural number n such that for every natural number m such that n m holds ∥(∑_{α=0}^{κ}(s_{1})(α))_{κ∈ℕ}(m) − (∑_{α=0}^{κ}(s_{1})(α))_{κ∈ℕ}(n)∥ < p.

(30) Let X be a real normed space, s be a sequence of X, and n, m be natural numbers. If n m, then ∥(∑_{α=0}^{κ} s(α))_{κ∈ℕ}(m) − (∑_{α=0}^{κ} s(α))_{κ∈ℕ}(n)∥ |(∑_{α=0}^{κ}∥s∥(α))_{κ∈ℕ}(m) − (∑_{α=0}^{κ}∥s∥(α))_{κ∈ℕ}(n)|.

(31) For every real Banach space X and for every sequence s_{1} of X such that s_{1} is norm-summable holds s_{1} is summable.

(32) Let X be a real normed space, r_{1} be a sequence of real numbers, and s_{5} be a sequence of X. Suppose r_{1} is summable and there exists a natural number m such that for every natural number n such that m n holds ∥s_{5}(n)∥ r_{1}(n). Then s_{5} is norm-summable.

(33) Let X be a real normed space and s_{2}, s_{5} be sequences of X. Suppose for every natural number n holds 0 ∥s_{2}∥(n) and ∥s_{2}∥(n) ∥s_{5}∥(n) and s_{5} is norm-summable. Then s_{2} is norm-summable and ∑∥s_{2}∥ ∑∥s_{5}∥.

(34) Let X be a real normed space and s_{1} be a sequence of X. Suppose that (i) for every natural number n holds ∥s_{1}∥(n) > 0, and (ii) there exists a natural number m such that for every natural number n

such that n m holds \frac{∥s_{1}∥(n+1)}{∥s_{1}∥(n)} 1. Then s_{1} is not norm-summable.

(35) Let X be a real normed space, s_{1} be a sequence of X, and r_{1} be a sequence of real numbers. Suppose for every natural number n holds r_{1}(n) = ^{n} \sqrt{∥s_{1}∥(n)} and r_{1} is convergent and limr_{1} < 1. Then s_{1} is normsummable.

(36) Let X be a real normed space, s_{1} be a sequence of X, and r_{1} be a

sequence of real numbers. Suppose that

(i) for every natural number n holds r_{1}(n) = ^{n} \sqrt{∥s_{1}∥(n)}, and (ii) there exists a natural number m such that for every natural number n

such that m n holds r_{1}(n) 1. Then ∥s_{1}∥ is not summable.

(37) Let X be a real normed space, s_{1} be a sequence of X, and r_{1} be a sequence of real numbers. Suppose for every natural number n holds r_{1}(n) = ^{n} \sqrt{∥s_{1}∥(n)} and r_{1} is convergent and limr_{1} > 1. Then s_{1} is not norm-summable.

(38) Let X be a real normed space, s_{1} be a sequence of X, and r_{1} be a sequence of real numbers. Suppose ∥s_{1}∥ is non-increasing and for every natural number n holds r_{1}(n) = 2^{n} · ∥s_{1}∥(2^{n}). Then s_{1} is norm-summable if and only if r_{1} is summable.

(39) Let X be a real normed space, s_{1} be a sequence of X, and p be a real number. Suppose p > 1 and for every natural number n such that n 1 holds ∥s_{1}∥(n) = \frac{1}{n^{p}} . Then s_{1} is norm-summable.

(40) Let X be a real normed space, s_{1} be a sequence of X, and p be a real number. Suppose p 1 and for every natural number n such that n 1 holds ∥s_{1}∥(n) = \frac{1}{n^{p}} . Then s_{1} is not norm-summable.

(41) Let X be a real normed space, s_{1} be a sequence of X, and r_{1} be a sequence of real numbers. Suppose for every natural number n holds s_{1}(n) ≠ 0_{X} and r_{1}(n) = \frac{∥s_{1}∥(n+1)}{∥s_{1}∥(n)} and r_{1} is convergent and limr_{1} < 1. Then s_{1} is norm-summable.

(42) Let X be a real normed space and s_{1} be a sequence of X. Suppose that (i) for every natural number n holds s_{1}(n) ≠ 0_{X}, and (ii) there exists a natural number m such that for every natural number n

such that n m holds \frac{∥s_{1}∥(n+1)}{∥s_{1}∥(n)} 1. Then s_{1} is not norm-summable.

Let X be a real Banach space. Observe that every sequence of X which is norm-summable is also summable.

2. Basic Properties of Sequences of Banach Algebra

The scheme ExNCBASeq deals with a non empty normed algebra structure A and a unary functor F yielding a point of A, and states that:

There exists a sequence S of A such that for every natural number n holds S(n) = F(n)

for all values of the parameters.

The following proposition is true

(43) Let X be a Banach algebra, x, y, z be elements of X, and a, b be real numbers. Then x+y = y+x and (x+y)+z = x+(y+z) and x+0_{X} = x and there exists an element t of X such that x+t = 0_{X} and (x·y)·z = x·(y ·z) and 1·x = x and 0·x = 0_{X} and a·0_{X} = 0_{X} and (− 1)·x = − x and x·1_{X} = x and 1_{X} · x = x and x · (y + z) = x · y + x · z and (y + z) · x = y · x + z · x and a·(x·y) = (a·x)·y and a·(x+y) = a·x+a·y and (a+b)·x = a·x+b·x and (a · b) · x = a · (b · x) and (a · b) · (x · y) = a · x · (b · y) and a · (x · y) = x · (a · y) and 0_{X} · x = 0_{X} and x · 0_{X} = 0_{X} and x · (y − z) = x · y − x · z and (y − z) · x = y · x − z · x and (x + y) − z = x + (y − z) and (x − y) + z = x − (y − z) and x − y − z = x − (y + z) and x + y = (x − z) + (z + y) and x − y = (x − z) + (z − y) and x = (x − y) + y and x = y − (y − x) and ∥x∥ = 0 iff x = 0_{X} and ∥a · x∥ = |a| · ∥x∥ and ∥x + y∥ ∥x∥ + ∥y∥ and ∥x · y∥ ∥x∥ · ∥y∥ and ∥1_{X}∥ = 1 and X is complete.

Let X be a non empty multiplicative loop structure and let v be an element of X. We say that v is invertible if and only if:

(Def. 8) There exists an element w of X such that v · w = 1_{X} and w · v = 1_{X}.

Let X be a non empty normed algebra structure, let S be a sequence of X, and let a be an element of X. The functor a · S yielding a sequence of X is defined by:

(Def. 9) For every natural number n holds (a · S)(n) = a · S(n).

Let X be a non empty normed algebra structure, let S be a sequence of X, and let a be an element of X. The functor S · a yields a sequence of X and is defined by:

(Def. 10) For every natural number n holds (S · a)(n) = S(n) · a.

Let X be a non empty normed algebra structure and let s_{2}, s_{5} be sequences of X. The functor s_{2} · s_{5} yielding a sequence of X is defined as follows:

the series on banach algebra 137

(Def. 11) For every natural number n holds (s_{2} · s_{5})(n) = s_{2}(n) · s_{5}(n). Let X be a Banach algebra and let x be an element of X. Let us assume that

x is invertible. The functor x^{−1} yielding an element of X is defined as follows: (Def. 12) x · x^{−1} = 1_{X} and x^{−1} · x = 1_{X}.

Let X be a Banach algebra and let z be an element of X. The functor (z^{κ})_{κ∈ℕ} yielding a sequence of X is defined as follows:

(Def. 13) (z^{κ})_{κ∈ℕ}(0) = 1_{X} and for every natural number n holds (z^{κ})_{κ∈ℕ}(n+1) = (z^{κ})_{κ∈ℕ}(n) · z.

Let X be a Banach algebra, let z be an element of X, and let n be a natural number. The functor z_{ℕ}^{n} yields an element of X and is defined by:

(Def. 14) z_{ℕ}^{n} = (z^{κ})_{κ∈ℕ}(n).

One can prove the following four propositions:

(44) For every Banach algebra X and for every element z of X holds z_{ℕ}^{0} = 1_{X}.

(45) For every Banach algebra X and for every element z of X such that ∥z∥ < 1 holds (z^{κ})_{κ∈ℕ} is summable and norm-summable.

(46) Let X be a Banach algebra and x be a point of X. If ∥1_{X} − x∥ < 1, then ((1_{X} − x)^{κ})_{κ∈ℕ} is summable and ((1_{X} − x)^{κ})_{κ∈ℕ} is norm-summable.

(47) For every Banach algebra X and for every point x of X such that ∥1_{X} − x∥ < 1 holds x is invertible and x^{−1} = ∑(((1_{X} − x)^{κ})_{κ∈ℕ}).

References

[1] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41–46, 1990.

[2] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91–96, 1990.

[3] Czesław Byliński. Binary operations. Formalized Mathematics, 1(1):175–180, 1990.

[4] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55– 65, 1990.

[5] Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153–164, 1990.

[6] Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35–40, 1990.

[7] Jarosław Kotowicz. Convergent sequences and the limit of sequences. Formalized Mathematics, 1(2):273–275, 1990.

[8] Jarosław Kotowicz. Monotone real sequences. Subsequences. Formalized Mathematics, 1(3):471–475, 1990.

[9] Jarosław Kotowicz. Real sequences and basic operations on them. Formalized Mathematics, 1(2):269–272, 1990.

[10] Eugeniusz Kusak, Wojciech Leończuk, and Michał Muzalewski. Abelian groups, fields and vector spaces. Formalized Mathematics, 1(2):335–342, 1990.

[11] Jan Popiołek. Some properties of functions modul and signum. Formalized Mathematics, 1(2):263–264, 1990.

[12] Jan Popiołek. Real normed space. Formalized Mathematics, 2(1):111–115, 1991.

[13] Konrad Raczkowski and Andrzej Nędzusiak. Real exponents and logarithms. Formalized Mathematics, 2(2):213–216, 1991.

[14] Konrad Raczkowski and Andrzej Nędzusiak. Series. Formalized Mathematics, 2(4):449– 452, 1991.

[15] Yasunari Shidama. Banach space of bounded linear operators. Formalized Mathematics, 12(1):39–48, 2003.

138 yasunari shidama

[16] Yasunari Shidama. The Banach algebra of bounded linear operators. Formalized Mathematics, 12(2):103–108, 2004.

[17] Yasumasa Suzuki, Noboru Endou, and Yasunari Shidama. Banach space of absolute summable real sequences. Formalized Mathematics, 11(4):377–380, 2003.

[18] Andrzej Trybulec. Subsets of complex numbers. To appear in Formalized Mathematics.

[19] Andrzej Trybulec. Tarski Grothendieck set theory. Formalized Mathematics, 1(1):9–11, 1990.

[20] Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291–296, 1990.

[21] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67–71, 1990.

[22] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73–83, 1990.

Formulas and Identities of Trigonometric Functions

Pacharapokin Chanapat

Shinshu University Nagano

Kanchun Shinshu University Nagano Hiroshi Yamazaki Shinshu University Nagano

Summary. In this article, we concentrated especially on addition formulas of fundamental trigonometric functions, and their identities.

MML Identifier: SIN COS4.

The articles [1] and [2] provide the notation and terminology for this paper. In this paper t_{1}, t_{2}, t_{3}, t_{4} denote real numbers. Let us consider t_{1}. The functor tant_{1} yielding a real number is defined by:

(Def. 1) tant_{1} = \begin{matrix}sin & t_{1} \\ cos & t_{1} \end{matrix} .

Let us consider t_{1}. The functor cott_{1} yields a real number and is defined by: (Def. 2) cott_{1} = \begin{matrix}cos & t_{1} \\ sin & t_{1} \end{matrix} .

Let us consider t_{1}. The functor cosect_{1} yielding a real number is defined as follows:

(Def. 3) cosect_{1} = \frac{1}{sin t_{1}} .

Let us consider t_{1}. The functor sect_{1} yielding a real number is defined by: (Def. 4) sect_{1} = \frac{1}{cos t_{1}} .

Next we state a number of propositions:

(1) tant_{1} = \frac{1}{cot t_{1}} .

(2) tan(− t_{1}) = − tant_{1}.

(3) cosec(− t_{1}) = − \frac{1}{sin t_{1}} .

(4) cot(− t_{1}) = − cott_{1}.

(5) If cost_{2} ≠ 0, then cost_{2} · sect_{2} = 1.

(6) sint_{1} · sint_{1} = 1 − cost_{1} · cost_{1}.

(7) cost_{1} · cost_{1} = 1 − sint_{1} · sint_{1}.

(8) If cost_{1} ≠ 0, then sint_{1} = cost_{1} · tant_{1}.

(9) sin(t_{2} − t_{3}) = sint_{2} · cost_{3} − cost_{2} · sint_{3}.

(10) cos(t_{2} − t_{3}) = cost_{2} · cost_{3} + sint_{2} · sint_{3}.

(11) If cost_{2} ≠ 0 and cost_{3} ≠ 0, then tan(t_{2} + t_{3}) = \frac{tan t_{2}+tan t_{3}}{1−tan t_{2}·tan t_{3}} .

(12) If cost_{2} ≠ 0 and cost_{3} ≠ 0, then tan(t_{2} − t_{3}) = \frac{tan t_{2}−tan t_{3}}{1+tan t_{2}·tan t_{3}}.

(13) If sint_{2} ≠ 0 and sint_{3} ≠ 0, then cot(t_{2} + t_{3}) = \frac{cot t_{2}·cot t_{3}−1}{cot t_{3}+cot t_{2}} .

(14) If sint_{2} ≠ 0 and sint_{3} ≠ 0, then cot(t_{2} − t_{3}) = \frac{cot t_{2}·cot t_{3}+1}{cot t_{3}−cot t_{2}}.

(15) If cost_{2} ≠ 0 and cost_{3} ≠ 0 and cost_{4} ≠ 0, then sin(t_{2} + t_{3} + t_{4}) = cost_{2} · cost_{3} · cost_{4} · ((tant_{2} + tant_{3} + tant_{4}) − tant_{2} · tant_{3} · tant_{4}).

(16) If cost_{2} ≠ 0 and cost_{3} ≠ 0 and cost_{4} ≠ 0, then cos(t_{2} + t_{3} + t_{4}) = cost_{2} · cost_{3} · cost_{4} · (1 − tant_{3} · tant_{4} − tant_{4} · tant_{2} − tant_{2} · tant_{3}).

(17) If cost_{2} ≠ 0 and cost_{3} ≠ 0 and cost_{4} ≠ 0, then tan(t_{2} + t_{3} + t_{4}) = \frac{(tan t_{2}+tan t_{3}+tan t_{4})−tan t_{2}·tan t_{3}·tan t_{4}}{1−tan t_{3}·tan t_{4}−tan t_{4}·tan t_{2}−tan t_{2}·tan t_{3}}.

(18) If sint_{2} ≠ 0 and sint_{3} ≠ 0 and sint_{4} ≠ 0, then cot(t_{2} + t_{3} + t_{4}) = \frac{cot t_{2}·cot t_{3}·cot t_{4}−cot t_{2}−cot t_{3}−cot t_{4}}{(cot t_{3}·cot t_{4}+cot t_{4}·cot t_{2}+cot t_{2}·cot t_{3})−1}.

(19) sint_{2} + sint_{3} = 2 · (cos(\frac{t_{2}−t_{3}}{2} ) · sin(\frac{t_{2}+t_{3}}{2} )).

(20) sint_{2} − sint_{3} = 2 · (cos(\frac{t_{2}+t_{3}}{2} ) · sin(\frac{t_{2}−t_{3}}{2} )).

(21) cost_{2} + cost_{3} = 2 · (cos(\frac{t_{2}+t_{3}}{2} ) · cos(\frac{t_{2}−t_{3}}{2} )).

(22) cost_{2} − cost_{3} = − 2 · (sin(\frac{t_{2}+t_{3}}{2} ) · sin(\frac{t_{2}−t_{3}}{2} )).

(23) If cost_{2} ≠ 0 and cost_{3} ≠ 0, then tant_{2} + tant_{3} = \frac{sin(t_{2}+t_{3})}{cos t_{2}·cos t_{3}}.

(24) If cost_{2} ≠ 0 and cost_{3} ≠ 0, then tant_{2} − tant_{3} = \frac{sin(t_{2}−t_{3})}{cos t_{2}·cos t_{3}}.

(25) If cost_{2} ≠ 0 and sint_{3} ≠ 0, then tant_{2} + cott_{3} = \frac{cos(t_{2}−t_{3})}{cos t_{2}·sin t_{3}}.

(26) If cost_{2} ≠ 0 and sint_{3} ≠ 0, then tant_{2} − cott_{3} = − \frac{cos(t_{2}+t_{3})}{cos t_{2}·sin t_{3}}.

(27) If sint_{2} ≠ 0 and sint_{3} ≠ 0, then cott_{2} + cott_{3} = \frac{sin(t_{2}+t_{3})}{sin t_{2}·sin t_{3}}.

(28) If sint_{2} ≠ 0 and sint_{3} ≠ 0, then cott_{2} − cott_{3} = − \frac{sin(t_{2}−t_{3})}{sin t_{2}·sin t_{3}}.

(29) sin(t_{2} + t_{3}) + sin(t_{2} − t_{3}) = 2 · (sint_{2} · cost_{3}).

(30) sin(t_{2} + t_{3}) − sin(t_{2} − t_{3}) = 2 · (cost_{2} · sint_{3}).

(31) cos(t_{2} + t_{3}) + cos(t_{2} − t_{3}) = 2 · (cost_{2} · cost_{3}).

(32) cos(t_{2} + t_{3}) − cos(t_{2} − t_{3}) = − 2 · (sint_{2} · sint_{3}).

(33) sint_{2} · sint_{3} = − \frac{1}{2} · (cos(t_{2} + t_{3}) − cos(t_{2} − t_{3})) .

(34) sint_{2} · cost_{3} = \frac{1}{2} · (sin(t_{2} + t_{3}) + sin(t_{2} − t_{3})).

(35) cost_{2} · sint_{3} = \frac{1}{2} · (sin(t_{2} + t_{3}) − sin(t_{2} − t_{3})).

(36) cost_{2} · cost_{3} = \frac{1}{2} · (cos(t_{2} + t_{3}) + cos(t_{2} − t_{3})).

(37) sint_{2} · sint_{3} · sint_{4} = \frac{1}{4} · ((sin((t_{2} + t_{3}) − t_{4}) + sin((t_{3} + t_{4}) − t_{2}) + sin((t_{4} + t_{2}) − t_{3})) − sin(t_{2} + t_{3} + t_{4})).

(38) sint_{2} · sint_{3} · cost_{4} = \frac{1}{4} · ((− cos((t_{2} + t_{3}) − t_{4}) + cos((t_{3} + t_{4}) − t_{2}) + cos((t_{4} + t_{2}) − t_{3})) − cos(t_{2} + t_{3} + t_{4})).

(39) sint_{2} ·cost_{3} ·cost_{4} = \frac{1}{4} ·((sin((t_{2} +t_{3})−t_{4})−sin((t_{3} +t_{4})−t_{2}))+sin((t_{4} + t_{2}) − t_{3}) + sin(t_{2} + t_{3} + t_{4})).

(40) cost_{2} ·cost_{3} ·cost_{4} = \frac{1}{4} ·(cos((t_{2} +t_{3})−t_{4})+cos((t_{3} +t_{4})−t_{2})+cos((t_{4} + t_{2}) − t_{3}) + cos(t_{2} + t_{3} + t_{4})).

(41) sin(t_{2} + t_{3}) · sin(t_{2} − t_{3}) = sint_{2} · sint_{2} − sint_{3} · sint_{3}.

(42) sin(t_{2} + t_{3}) · sin(t_{2} − t_{3}) = cost_{3} · cost_{3} − cost_{2} · cost_{2}.

(43) sin(t_{2} + t_{3}) · cos(t_{2} − t_{3}) = sint_{2} · cost_{2} + sint_{3} · cost_{3}.

(44) cos(t_{2} + t_{3}) · sin(t_{2} − t_{3}) = sint_{2} · cost_{2} − sint_{3} · cost_{3}.

(45) cos(t_{2} + t_{3}) · cos(t_{2} − t_{3}) = cost_{2} · cost_{2} − sint_{3} · sint_{3}.

(46) cos(t_{2} + t_{3}) · cos(t_{2} − t_{3}) = cost_{3} · cost_{3} − sint_{2} · sint_{2}.

(47) If cost_{2} ≠ 0 and cost_{3} ≠ 0, then \frac{sin(t_{2}+t_{3})}{sin(t_{2}−t_{3})} = \frac{tan t_{2}+tan t_{3}}{tan t_{2}−tan t_{3}}.

(48) If cost_{2} ≠ 0 and cost_{3} ≠ 0, then \frac{cos(t_{2}+t_{3})}{cos(t_{2}−t_{3})} = \frac{1−tan t_{2}·tan t_{3}}{1+tan t_{2}·tan t_{3}}.

(49) \frac{sin t_{2}+sin t_{3}}{sin t_{2}−sin t_{3}} = tan(\frac{t_{2}+t_{3}}{2} ) · cot(\frac{t_{2}−t_{3}}{2} ).

(50) If cos(\frac{t_{2}−t_{3}}{2} ) ≠ 0, then \frac{sin t_{2}+sin t_{3}}{cos t_{2}+cos t_{3}} = tan(\frac{t_{2}+t_{3}}{2} ).

(51) If cos(\frac{t_{2}+t_{3}}{2} ) ≠ 0, then \frac{sin t_{2}−sin t_{3}}{cos t_{2}+cos t_{3}} = tan(\frac{t_{2}−t_{3}}{2} ).

(52) If sin(\frac{t_{2}+t_{3}}{2} ) ≠ 0, then \frac{sin t_{2}+sin t_{3}}{cos t_{3}−cos t_{2}} = cot(\frac{t_{2}−t_{3}}{2} ).

(53) If sin(\frac{t_{2}−t_{3}}{2} ) ≠ 0, then \frac{sin t_{2}−sin t_{3}}{cos t_{3}−cos t_{2}} = cot(\frac{t_{2}+t_{3}}{2} ).

(54) \frac{cos t_{2}+cos t_{3}}{cos t_{2}−cos t_{3}} = cot(\frac{t_{2}+t_{3}}{2} ) · cot(\frac{t_{3}−t_{2}}{2} ).

References

[1] Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35–40, 1990.

[2] Yuguang Yang and Yasunari Shidama. Trigonometric functions and existence of circle ratio. Formalized Mathematics, 7(2):255–263, 1998.

The Class of Series-Parallel Graphs. Part III

Krzysztof Retel University of Bialystok

Summary. This paper contains some facts and theorems relating to the following operations on graphs: union, sum, complement and “embeds”. We also introduce connected graphs to prove that a finite irreflexive symmetric N-free graph is a finite series-parallel graph. This article continues the formalization of [22].

MML Identifier: NECKLA 3.

The papers [25], [24], [28], [12], [29], [31], [30], [2], [13], [1], [27], [18], [17], [8], [14], [16], [20], [23], [7], [10], [26], [11], [4], [6], [19], [15], [5], [21], [3], and [9] provide the notation and terminology for this paper.

1. Preliminaries

In this paper A, B, a, b, c, d, e, f, g, h denote sets.

One can prove the following three propositions:

(1) id_{A}↾B = id_{A} ∩ [: B, B:].

(2) id_{{a,b,c,d}} = {⟨⟨a, a⟩⟩, ⟨⟨b, b⟩⟩, ⟨⟨c, c⟩⟩, ⟨⟨d, d⟩⟩}.

(3) [: {a, b, c, d}, {e, f, g, h}:] = {⟨⟨a, e⟩⟩, ⟨⟨a, f⟩⟩, ⟨⟨b, e⟩⟩, ⟨⟨b, f⟩⟩, ⟨⟨a, g⟩⟩, ⟨⟨a, h⟩⟩, ⟨⟨b, g⟩⟩, ⟨⟨b, h⟩⟩} ∪ {⟨⟨c, e⟩⟩, ⟨⟨c, f⟩⟩, ⟨⟨d, e⟩⟩, ⟨⟨d, f⟩⟩, ⟨⟨c, g⟩⟩, ⟨⟨c, h⟩⟩, ⟨⟨d, g⟩⟩, ⟨⟨d, h⟩⟩}.

Let X, Y be trivial sets. Observe that every relation between X and Y is trivial.

We now state the proposition

(4) For every trivial set X and for every binary relation R on X such that R is non empty there exists a set x such that R = {⟨⟨x, x⟩⟩}.

Let X be a trivial set. Observe that every binary relation on X is trivial, reflexive, symmetric, transitive, and strongly connected.

We now state the proposition

(5) For every non empty trivial set X holds every binary relation on X is symmetric in X.

One can verify that there exists a relational structure which is non empty, strict, finite, irreflexive, and symmetric.

Let L be an irreflexive relational structure. Observe that every full relational substructure of L is irreflexive.

Let L be a symmetric relational structure. Note that every full relational substructure of L is symmetric.

One can prove the following proposition

(6) Let R be an irreflexive symmetric relational structure. Suppose \overline{\overline{the carrier of R}} = 2. Then there exist sets a, b such that the carrier of R = {a, b} but the internal relation of R = {⟨⟨a, b⟩⟩, ⟨⟨b, a⟩⟩} or the internal relation of R = ∅.

2. Some Facts about Operations “UnionOf” and “SumOf”

Let R be a non empty relational structure and let S be a relational structure. Note that UnionOf(R, S) is non empty and SumOf(R, S) is non empty.

Let R be a relational structure and let S be a non empty relational structure. Observe that UnionOf(R, S) is non empty and SumOf(R, S) is non empty.

Let R, S be finite relational structures. One can check that UnionOf(R, S) is finite and SumOf(R, S) is finite.

Let R, S be symmetric relational structures. One can check that UnionOf(R, S) is symmetric and SumOf(R, S) is symmetric.

Let R, S be irreflexive relational structures. Observe that UnionOf(R, S) is irreflexive.

The following four propositions are true:

(7) Let R, S be irreflexive relational structures. Suppose the carrier of R misses the carrier of S. Then SumOf(R, S) is irreflexive.

(8) For all relational structures R_{1}, R_{2} holds UnionOf(R_{1}, R_{2}) = UnionOf(R_{2}, R_{1}) and SumOf(R_{1}, R_{2}) = SumOf(R_{2}, R_{1}).

(9) Let G be an irreflexive relational structure and G_{1}, G_{2} be relational structures. If G = UnionOf(G_{1}, G_{2}) or G = SumOf(G_{1}, G_{2}), then G_{1} is irreflexive and G_{2} is irreflexive.

(10) Let G be a non empty relational structure and H_{1}, H_{2} be relational

structures. Suppose that

(i) the carrier of H_{1} misses the carrier of H_{2}, and (ii) the relational structure of G = UnionOf(H_{1}, H_{2}) or the relational struc-

ture of G = SumOf(H_{1}, H_{2}). Then H_{1} is a full relational substructure of G and H_{2} is a full relational substructure of G.

3. Theorems Relating to the Complement of Relational Structure

One can prove the following proposition

(11) The internal relation of ComplRelStrNecklace4 = {⟨⟨0, 2⟩⟩, ⟨⟨2, 0⟩⟩, ⟨⟨0,

3⟩⟩, ⟨⟨3, 0⟩⟩, ⟨⟨1, 3⟩⟩, ⟨⟨3, 1⟩⟩}. Let R be a relational structure. Note that ComplRelStrR is irreflexive. Let R be a symmetric relational structure. Note that ComplRelStrR is sym-

metric.

Next we state several propositions:

(12) For every relational structure R holds the internal relation of R misses the internal relation of ComplRelStrR.

(13) For every relational structure R holds id_{the carrier of R} misses the internal relation of ComplRelStrR.

(14) Let G be a relational structure. Then [: the carrier of G, the carrier of G:] = id_{the carrier of G} ∪ the internal relation of G ∪ the internal relation of ComplRelStrG.

(15) For every strict irreflexive relational structure G such that G is trivial holds ComplRelStrG = G.

(16) For every strict irreflexive relational structure G holds ComplRelStrComplRelStrG = G.

(17) For all relational structures G_{1}, G_{2} such that the carrier of G_{1} misses the carrier of G_{2} holds ComplRelStrUnionOf(G_{1}, G_{2}) = SumOf(ComplRelStrG_{1}, ComplRelStrG_{2}).

(18) For all relational structures G_{1}, G_{2} such that the carrier of G_{1} misses the carrier of G_{2} holds ComplRelStrSumOf(G_{1}, G_{2}) = UnionOf(ComplRelStrG_{1}, ComplRelStrG_{2}).

(19) Let G be a relational structure and H be a full relational substructure of G. Then the internal relation of ComplRelStrH = (the internal relation of ComplRelStrG) |^{2} (the carrier of ComplRelStrH).

(20) Let G be a non empty irreflexive relational structure, x be an element of the carrier of G, and x′ be an element of the carrier of ComplRelStrG. If x = x′, then ComplRelStrsub(Ω_{G} \ {x}) = sub(Ω_{ComplRelStr G} \ {x′}).

4. Another Facts Relating to Operation “embeds”

Let us observe that every non empty relational structure which is trivial and strict is also N-free.

The following propositions are true:

(21) Let R be a reflexive antisymmetric relational structure and S be a relational structure. Then there exists a map f from R into S such that for all elements x, y of the carrier of R holds ⟨⟨x, y⟩⟩∈ the internal relation of R iff ⟨⟨f(x), f(y)⟩⟩∈ the internal relation of S if and only if S embeds R.

(22) Let G be a non empty relational structure and H be a non empty full relational substructure of G. Then G embeds H.

(23) Let G be a non empty relational structure and H be a non empty full relational substructure of G. If G is N-free, then H is N-free.

(24) For every non empty irreflexive relational structure G holds G embeds Necklace4 iff ComplRelStrG embeds Necklace4.

(25) For every non empty irreflexive relational structure G holds G is N-free iff ComplRelStrG is N-free.

5. Connected Graphs

Let R be a relational structure. A path of R is a reduction sequence w.r.t. the internal relation of R.

Let R be a relational structure. We say that R is path-connected if and only if the condition (Def. 1) is satisfied.

(Def. 1) Let x, y be sets. Suppose x ∈ the carrier of R and y ∈ the carrier of R and x ≠ y. Then the internal relation of R reduces x to y or the internal relation of R reduces y to x.

One can check that every relational structure which is empty is also pathconnected.

One can check that every non empty relational structure which is connected is also path-connected.

We now state the proposition

(26) Let R be a non empty transitive reflexive relational structure and x, y be elements of R. Suppose the internal relation of R reduces x to y. Then ⟨⟨x, y⟩⟩∈ the internal relation of R.

One can check that every non empty transitive reflexive relational structure which is path-connected is also connected.

Next we state the proposition (27) Let R be a symmetric relational structure and x, y be sets. Suppose

x ∈ the carrier of R and y ∈ the carrier of R. Suppose the internal relation of R reduces x to y. Then the internal relation of R reduces y to x.

Let R be a symmetric relational structure. Let us observe that R is pathconnected if and only if the condition (Def. 2) is satisfied.

(Def. 2) Let x, y be sets. Suppose x ∈ the carrier of R and y ∈ the carrier of R

and x ≠ y. Then the internal relation of R reduces x to y. Let R be a relational structure and let x be an element of R. The functor

component(x) yielding a subset of R is defined as follows: (Def. 3) component(x) = [x]_{EqCl(the internal relation of R)}.

Next we state the proposition

(28) For every non empty relational structure R and for every element x of

R holds x ∈ component(x). Let R be a non empty relational structure and let x be an element of R.

Note that component(x) is non empty.

Next we state a number of propositions:

(29) Let R be a relational structure, x be an element of R, and y be a set. If y ∈ component(x), then ⟨⟨x, y⟩⟩∈ EqCl(the internal relation of R).

(30) Let R be a relational structure, x be an element of R, and A be a set. Then A = component(x) if and only if for every set y holds y ∈ A iff ⟨⟨x, y⟩⟩∈ EqCl(the internal relation of R).

(31) Let R be a non empty irreflexive symmetric relational structure. Suppose R is not path-connected. Then there exist non empty strict irreflexive symmetric relational structures G_{1}, G_{2} such that the carrier of G_{1} misses the carrier of G_{2} and the relational structure of R = UnionOf(G_{1}, G_{2}).

(32) Let R be a non empty irreflexive symmetric relational structure. Suppose ComplRelStrR is not path-connected. Then there exist non empty strict irreflexive symmetric relational structures G_{1}, G_{2} such that the carrier of G_{1} misses the carrier of G_{2} and the relational structure of R = SumOf(G_{1}, G_{2}).

(33) For every irreflexive relational structure G such that G ∈ FinRelStrSp holds ComplRelStrG ∈ FinRelStrSp.

(34) Let R be an irreflexive symmetric relational structure. Suppose \overline{\overline{the carrier of R}} = 2 and the carrier of R ∈ U_{0}. Then the relational structure of R ∈ FinRelStrSp.

(35) For every relational structure R such that R ∈ FinRelStrSp holds R is symmetric.

(36) Let G be a relational structure, H_{1}, H_{2} be non empty relational structures, x be an element of the carrier of H_{1}, and y be an element of the carrier of H_{2}. Suppose G = UnionOf(H_{1}, H_{2}) and the carrier of H_{1} misses the carrier of H_{2}. Then ⟨⟨x, y⟩⟩ ∉ the internal relation of G.

(37) Let G be a relational structure, H_{1}, H_{2} be non empty relational structures, x be an element of the carrier of H_{1}, and y be an element of the carrier of H_{2}. If G = SumOf(H_{1}, H_{2}), then ⟨⟨x, y⟩⟩ ∉ the internal relation of ComplRelStrG.

(38) Let G be a non empty symmetric relational structure, x be an element of the carrier of G, and R_{1}, R_{2} be non empty relational structures. Suppose the carrier of R_{1} misses the carrier of R_{2} and sub(Ω_{G} \ {x}) = UnionOf(R_{1}, R_{2}) and G is path-connected. Then there exists an element b of the carrier of R_{1} such that ⟨⟨b, x⟩⟩∈ the internal relation of G.

(39) Let G be a non empty symmetric irreflexive relational structure, a, b, c, d be elements of the carrier of G, and Z be a subset of the carrier of G. Suppose that Z = {a, b, c, d} and a, b, c, d are mutually different and ⟨⟨a, b⟩⟩∈ the internal relation of G and ⟨⟨b, c⟩⟩∈ the internal relation of G and ⟨⟨c, d⟩⟩ ∈ the internal relation of G and ⟨⟨a, c⟩⟩ ∉ the internal relation of G and ⟨⟨a, d⟩⟩ ∉ the internal relation of G and ⟨⟨b, d⟩⟩ ∉ the internal relation of G. Then sub(Z) embeds Necklace4.

(40) Let G be a non empty irreflexive symmetric relational structure, x be an element of the carrier of G, and R_{1}, R_{2} be non empty relational structures. Suppose that

(i) the carrier of R_{1} misses the carrier of R_{2}, (ii) sub(Ω_{G} \ {x}) = UnionOf(R_{1}, R_{2}), (iii) G is non trivial and path-connected, and (iv) ComplRelStrG is path-connected.

Then G embeds Necklace4.

(41) Let G be a non empty strict finite irreflexive symmetric relational structure. Suppose G is N-free and the carrier of G ∈ U_{0}. Then the relational structure of G ∈ FinRelStrSp.

Acknowledgments

The author would like to thank Professor Grzegorz Bancerek for many helpful conversations during the preparation of this paper, which allowed me to publish it and for aquainting me with new search engine in MML (MML Query). I am also very grateful to Professor Andrzej Trybulec, Adam Naumowicz, and Adam Grabowski for their valuable advice.

References

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the class of series-parallel graphs. … 149

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[6] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107–114, 1990.

[7] Józef Białas. Group and field definitions. Formalized Mathematics, 1(3):433–439, 1990.

[8] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55– 65, 1990.

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[11] Czesław Byliński. Partial functions. Formalized Mathematics, 1(2):357–367, 1990.

[12] Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47–53, 1990.

[13] Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165–167, 1990.

[14] Wojciech Leończuk and Krzysztof Prażmowski. Incidence projective spaces. Formalized Mathematics, 2(2):225–232, 1991.

[15] Robert Milewski. Lattice of congruences in many sorted algebra. Formalized Mathematics, 5(4):479–483, 1996.

[16] Bogdan Nowak and Grzegorz Bancerek. Universal classes. Formalized Mathematics, 1(3):595–600, 1990.

[17] Beata Padlewska. Families of sets. Formalized Mathematics, 1(1):147–152, 1990.

[18] Beata Padlewska and Agata Darmochwał. Topological spaces and continuous functions. Formalized Mathematics, 1(1):223–230, 1990.

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[21] Krzysztof Retel. The class of series-parallel graphs. Part II. Formalized Mathematics, 11(3):289–291, 2003.

[22] Stephan Thomasse. On better-quasi-ordering countable series-parallel orders. Transactions of the American Mathematical Society, 352(6):2491–2505, 2000.

[23] Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics, 1(2):329–334, 1990.

[24] Andrzej Trybulec. Enumerated sets. Formalized Mathematics, 1(1):25–34, 1990.

[25] Andrzej Trybulec. Tarski Grothendieck set theory. Formalized Mathematics, 1(1):9–11, 1990.

[26] Wojciech A. Trybulec. Groups. Formalized Mathematics, 1(5):821–827, 1990.

[27] Wojciech A. Trybulec. Partially ordered sets. Formalized Mathematics, 1(2):313–319, 1990.

[28] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67–71, 1990.

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[31] Edmund Woronowicz and Anna Zalewska. Properties of binary relations. Formalized Mathematics, 1(1):85–89, 1990.

Relocability for SCM over Ring

Artur Korniłowicz^{1} University of Białystok Yasunari Shidama Shinshu University Nagano

MML Identifier: SCMRING4.

The notation and terminology used in this paper have been introduced in the following articles: [23], [27], [3], [4], [10], [28], [21], [7], [8], [5], [22], [1], [26], [6], [9], [19], [2], [15], [18], [16], [17], [24], [20], [12], [11], [25], [13], and [14].

1. On the Standard Computers

For simplicity, we use the following convention: i, j, k denote natural numbers, n denotes a natural number, N denotes a set with non empty elements, S denotes a standard IC-Ins-separated definite non empty non void AMI over N, l denotes an instruction-location of S, and f denotes a finite partial state of S.

Next we state the proposition

(1) ℕ ≈ the instruction locations of S.

Let us consider N, S. Observe that the instruction locations of S is infinite.

We now state the proposition

(2) il_{S}(i) + j = il_{S}(i + j).

Let N be a set with non empty elements, let S be a standard IC-Ins-separated definite non empty non void AMI over N, let l_{1} be an instruction-location of S, and let k be a natural number. The functor l_{1} − ′k yields an instruction-location of S and is defined as follows:

(Def. 1) l_{1} − ′k = il_{S}(locnum(l_{1}) − ′k). We now state a number of propositions:

(3) l − ′0 = l.

1The paper was written during the post-doctoral fellowship granted by Shinshu University, Japan.

(4) locnum(l) − ′k = locnum(l − ′k).

(5) (l + k) − ′k = l.

(6) il_{S}(i) − ′j = il_{S}(i − ′j).

(7) Let S be an IC-Ins-separated definite non empty non void AMI over N and p be a finite partial state of S. Then domDataPart(p) ⊆ (the carrier of S) \ ({IC_{S}} ∪ the instruction locations of S).

(8) Let S be an IC-Ins-separated definite realistic non empty non void AMI over N and p be a finite partial state of S. Then p is data-only if and only if domp ⊆ (the carrier of S) \ ({IC_{S}} ∪ the instruction locations of S).

(9) For all instruction-locations l_{2}, l_{3} of S holds Start-At(l_{2} + k) = Start-At(l_{3} + k) iff Start-At(l_{2}) = Start-At(l_{3}).

(10) For all instruction-locations l_{2}, l_{3} of S such that Start-At(l_{2}) = Start-At(l_{3}) holds Start-At(l_{2} − ′k) = Start-At(l_{3} − ′k).

(11) If l ∈ domf, then (Shift(f, k))(l + k) = f(l).

(12) domShift(f, k) = {i_{1} + k; i_{1} ranges over instruction-locations of S: i_{1} ∈ domf}.

(13) Let S be an Exec-preserving IC-Ins-separated definite realistic steadyprogrammed non empty non void AMI over N, s be a state of S, i be an instruction of S, and p be a programmed finite partial state of S. Then Exec(i, s+·p) = Exec(i, s)+·p.

2. SCM(R)

For simplicity, we follow the rules: R denotes a good ring, a, b denote Data-Locations of R, l_{1} denotes an instruction-location of SCM (R), I denotes an instruction of SCM (R), p denotes a finite partial state of SCM (R), s, s_{1}, s_{2} denote states of SCM (R), and q denotes a finite partial state of SCM .

One can prove the following propositions:

(14) The carrier of SCM (R) = {IC_{SCM (R)}} ∪ Data-Loc_{SCM} ∪ Instr-Loc_{SCM}.

(15) ObjectKind(l_{1}) = Instr_{SCM}(R).

(16) dl_{R}(n) = 2 · n + 1.

(17) il_{SCM (R)}(k) = 2 · k + 2.

(18) For every Data-Location d_{1} of R there exists a natural number i such that d_{1} = dl_{R}(i).

(19) For all natural numbers i, j such that i ≠ j holds dl_{R}(i) ≠ dl_{R}(j).

(20) a ≠ l_{1}.

(21) Data-Loc_{SCM} ⊆ doms.

(22) dom(s↾Data-Loc_{SCM}) = Data-Loc_{SCM}.

(23) If p = q, then DataPart(p) = DataPart(q).

(24) DataPart(p) = p↾Data-Loc_{SCM}.

(25) p is data-only iff domp ⊆ Data-Loc_{SCM}.

(26) domDataPart(p) ⊆ Data-Loc_{SCM}.

(27) Instr-Loc_{SCM} ⊆ doms.

(28) If p = q, then ProgramPart(p) = ProgramPart(q).

(29) domProgramPart(p) ⊆ Instr-Loc_{SCM}.

Let us consider R and let I be an element of the instructions of SCM (R). Observe that InsCode(I) is natural.

Next we state several propositions:

(30) InsCode(I) 7.

(31) IncAddr(goto l_{1}, k) = goto (l_{1} + k).

(32) IncAddr(if a = 0 goto l_{1}, k) = if a = 0 goto l_{1} + k.

(33) s(a) = (s+·Start-At(l_{1}))(a).

(34) Suppose IC_{(s_{1})} = IC_{(s_{2})} and for every Data-Location a of R holds s_{1}(a) = s_{2}(a) and for every instruction-location i of SCM (R) holds s_{1}(i) = s_{2}(i). Then s_{1} = s_{2}.

(35) Exec(IncAddr(CurInstr(s), k), s+·Start-At(IC_{s} + k)) = Following(s)+·Start-At(IC_{Following (s)} + k).

(36) If IC_{s} = il_{SCM (R)}(j + k), then Exec(I, s+·Start-At(IC_{s} − ′ k)) =

Exec(IncAddr(I, k), s)+·Start-At(IC_{Exec(IncAddr (I,k),s)} − ′k). Let us consider R. One can check that there exists a finite partial state of

SCM (R) which is autonomic and non programmed.

Let us consider R, let a be a Data-Location of R, and let r be an element of the carrier of R. Then a⟼→ . r is a finite partial state of SCM (R).

We now state a number of propositions:

(37) If R is non trivial, then for every autonomic finite partial state p of SCM (R) such that DataPart(p) ≠ ∅ holds IC_{SCM (R)} ∈ domp.

(38) If R is non trivial, then for every autonomic non programmed finite partial state p of SCM (R) holds IC_{SCM (R)} ∈ domp.

(39) For every autonomic finite partial state p of SCM (R) such that IC_{SCM (R)} ∈ domp holds IC_{p} ∈ domp.

(40) Suppose R is non trivial. Let p be an autonomic non programmed finite partial state of SCM (R). If p ⊆ s, then IC_{(Computation (s))(n)} ∈ domProgramPart(p).

(41) Suppose R is non trivial. Let p be an autonomic non programmed finite partial state of SCM (R). If p ⊆ s_{1} and p ⊆ s_{2}, then IC_{(Computation (s_{1}))(n)} = IC_{(Computation (s_{2}))(n)} and CurInstr((Computation(s_{1}))(n)) = CurInstr((Computation(s_{2}))(n)).

(42) Suppose R is non trivial. Let p be an autonomic non programmed finite partial state of SCM (R). If p ⊆ s_{1} and p ⊆ s_{2} and CurInstr((Computation(s_{1}))(n)) = a:=b and a ∈ domp, then (Computation(s_{1}))(n)(b) = (Computation(s_{2}))(n)(b).

(43) Suppose R is non trivial. Let p be an autonomic non programmed finite partial state of SCM (R). Suppose p ⊆ s_{1} and p ⊆ s_{2} and CurInstr((Computation(s_{1}))(n)) = AddTo(a, b) and a ∈ domp. Then (Computation(s_{1}))(n)(a) + (Computation(s_{1}))(n)(b) = (Computation(s_{2}))(n)(a) + (Computation(s_{2}))(n)(b).

(44) Suppose R is non trivial. Let p be an autonomic non programmed finite partial state of SCM (R). Suppose p ⊆ s_{1} and p ⊆ s_{2} and CurInstr((Computation(s_{1}))(n)) = SubFrom(a, b) and a ∈ domp. Then (Computation(s_{1}))(n)(a) − (Computation(s_{1}))(n)(b) = (Computation(s_{2}))(n)(a) − (Computation(s_{2}))(n)(b).

(45) Suppose R is non trivial. Let p be an autonomic non programmed finite partial state of SCM (R). Suppose p ⊆ s_{1} and p ⊆ s_{2} and CurInstr((Computation(s_{1}))(n)) = MultBy(a, b) and a ∈ domp. Then (Computation(s_{1}))(n)(a) · (Computation(s_{1}))(n)(b) = (Computation(s_{2}))(n)(a) · (Computation(s_{2}))(n)(b).

(46) Suppose R is non trivial. Let p be an autonomic non programmed finite partial state of SCM (R). Suppose p ⊆ s_{1} and p ⊆ s_{2} and CurInstr((Computation(s_{1}))(n)) = if a = 0 goto l_{1} and l_{1} ≠ Next(IC_{(Computation (s_{1}))(n)}). Then (Computation(s_{1}))(n)(a) = 0_{R} if and only if (Computation(s_{2}))(n)(a) = 0_{R}.

3. Relocability

Let N be a set with non empty elements, let S be a regular standard IC-Ins-separated definite non empty non void AMI over N, let k be a natural number, and let p be a finite partial state of S. The functor Relocated(p, k) yielding a finite partial state of S is defined as follows:

(Def. 2) Relocated(p, k) = Start-At(IC_{p} + k)+·IncAddr(Shift(ProgramPart(p),

k), k)+·DataPart(p). In the sequel S denotes a regular standard IC-Ins-separated definite non

empty non void AMI over N, g denotes a finite partial state of S, and i_{1} denotes an instruction-location of S.

One can prove the following propositions:

(47) DataPart(Relocated(g, k)) = DataPart(g).

(48) If S is realistic, then ProgramPart(Relocated(g, k)) = IncAddr(Shift(ProgramPart(g), k), k).

(49) If S is realistic, then domProgramPart(Relocated(g, k)) = {il_{S}(j + k); j ranges over natural numbers: il_{S}(j) ∈ domProgramPart(g)}.

(50) If S is realistic, then i_{1} ∈ domg iff i_{1} + k ∈ domRelocated(g, k).

(51) IC_{S} ∈ domRelocated(g, k).

(52) If S is realistic, then IC_{Relocated (g,k)} = IC_{g} + k.

(53) Let p be a programmed finite partial state of S and l be an instructionlocation of S. If l ∈ domp, then (IncAddr(p, k))(l) = IncAddr(π_{l}p, k).

(54) For every programmed finite partial state p of S holds Shift(IncAddr(p, i), i) = IncAddr(Shift(p, i), i).

(55) If S is realistic, then for every instruction I of S such that i_{1} ∈ domProgramPart(g) and I = g(i_{1}) holds IncAddr(I, k) = (Relocated(g, k))(i_{1} + k).

(56) If S is realistic, then Start-At(IC_{g} + k) ⊆ Relocated(g, k).

(57) If S is realistic, then for every data-only finite partial state q of S such that IC_{S} ∈ domg holds Relocated(g+·q, k) = Relocated(g, k)+·q.

(58) For every autonomic finite partial state p of SCM (R) such that p ⊆ s_{1} and Relocated(p, k) ⊆ s_{2} holds p ⊆ s_{1}+·s_{2}↾Data-Loc_{SCM}.

(59) Suppose R is non trivial. Let p be an autonomic finite partial state of SCM (R). Suppose IC_{SCM (R)} ∈ domp and p ⊆ s_{1} and Relocated(p, k) ⊆ s_{2} and s = s_{1}+·s_{2}↾Data-Loc_{SCM}. Let i be a natural number. Then IC_{(Computation (s_{1}))(i)} + k = IC_{(Computation (s_{2}))(i)} and IncAddr(CurInstr((Computation(s_{1}))(i)), k) = CurInstr((Computation(s_{2}))(i)) and (Computation(s_{1}))(i)↾ domDataPart (p) = (Computation(s_{2}))(i)↾ domDataPart(Relocated(p, k)) and (Computation(s))(i)↾Data-Loc_{SCM} = (Computation(s_{2}))(i)↾Data-Loc_{SCM}.

(60) Suppose R is non trivial. Let p be an autonomic finite partial state of SCM (R). If IC_{SCM (R)} ∈ domp, then p is halting iff Relocated(p, k) is halting.

(61) Suppose R is non trivial. Let p be an autonomic finite partial state of SCM (R). Suppose IC_{SCM (R)} ∈ domp and p ⊆ s. Let i be a natural number. Then (Computation(s+·Relocated(p, k)))(i) = (Computation(s))(i)+·Start-At(IC_{(Computation (s))(i)} + k)+·ProgramPart (Relocated(p, k)).

(62) Suppose R is non trivial. Let p be an autonomic finite partial state of SCM (R). Suppose IC_{SCM (R)} ∈ domp and Relocated(p, k) ⊆ s. Let i be a natural number. Then (Computation(s))(i) = (Computation(s+·p))(i)+·Start-At(IC_{(Computation (s+·p))(i)} + k)+·s↾ dom ProgramPart(p)+·ProgramPart(Relocated(p, k)).

(63) Suppose R is non trivial and IC_{SCM (R)} ∈ domp and p ⊆ s and Relocated(p, k) is autonomic. Let i be a natural number. Then 156 artur korniłowicz and yasunari shidama

(Computation(s))(i) = (Computation(s+·Relocated(p, k)))(i)+·Start-At (IC_{(Computation (s+·Relocated (p,k)))(i)}− ′k)+·s↾ domProgramPart(Relocated(p, k))+·ProgramPart(p).

(64) If R is non trivial and IC_{SCM (R)} ∈ domp, then p is autonomic iff Relocated(p, k) is autonomic.

(65) Suppose R is non trivial. Let p be a halting autonomic finite partial state of SCM (R). If IC_{SCM (R)} ∈ domp, then DataPart(Result(p)) = DataPart(Result(Relocated(p, k))).

(66) Suppose R is non trivial. Let F be a partial function from FinPartSt(SCM (R)) to FinPartSt(SCM (R)). Suppose IC_{SCM (R)} ∈ domp and F is data-only. Then p computes F if and only if Relocated(p, k) computes F.

References

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[8] Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153–164, 1990.

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Convergent Sequences in Complex Unitary Space

Noboru Endou Gifu National College of Technology

Summary. In this article, we introduce the notion of convergence sequence in complex unitary space and complex Hilbert space.

MML Identifier: CLVECT 2.

The terminology and notation used in this paper are introduced in the following papers: [15], [2], [14], [7], [1], [17], [3], [4], [10], [9], [16], [13], [11], [12], [8], [5], and [6].

1. Convergence in Complex Unitary Space

For simplicity, we adopt the following convention: X is a complex unitary space, x, y, w, g, g_{1}, g_{2} are points of X, z is a Complex, q, r, M are real numbers, s_{1}, s_{2}, s_{3}, s_{4} are sequences of X, k, n, m are natural numbers, and N_{1} is an increasing sequence of naturals.

Let us consider X, s_{1}. We say that s_{1} is convergent if and only if: (Def. 1) There exists g such that for every r such that r > 0 there exists m such

that for every n such that n m holds ρ(s_{1}(n), g) < r. Next we state several propositions:

(1) If s_{1} is constant, then s_{1} is convergent.

(2) If s_{2} is convergent and there exists k such that for every n such that k n holds s_{3}(n) = s_{2}(n), then s_{3} is convergent.

(3) If s_{2} is convergent and s_{3} is convergent, then s_{2} + s_{3} is convergent.

(4) If s_{2} is convergent and s_{3} is convergent, then s_{2} − s_{3} is convergent.

(5) If s_{1} is convergent, then z · s_{1} is convergent.

(6) If s_{1} is convergent, then − s_{1} is convergent.

(7) If s_{1} is convergent, then s_{1} + x is convergent.

(8) If s_{1} is convergent, then s_{1} − x is convergent.

(9) s_{1} is convergent if and only if there exists g such that for every r such that r > 0 there exists m such that for every n such that n m holds ∥s_{1}(n) − g∥ < r.

Let us consider X, s_{1}. Let us assume that s_{1} is convergent. The functor lims_{1} yields a point of X and is defined as follows:

(Def. 2) For every r such that r > 0 there exists m such that for every n such

that n m holds ρ(s_{1}(n), lims_{1}) < r. One can prove the following propositions:

(10) If s_{1} is constant and x ∈ rngs_{1}, then lims_{1} = x.

(11) If s_{1} is constant and there exists n such that s_{1}(n) = x, then lims_{1} = x.

(12) If s_{2} is convergent and there exists k such that for every n such that n k holds s_{3}(n) = s_{2}(n), then lims_{2} = lims_{3}.

(13) If s_{2} is convergent and s_{3} is convergent, then lim(s_{2}+s_{3}) = lims_{2}+lims_{3}.

(14) If s_{2} is convergent and s_{3} is convergent, then lim(s_{2}−s_{3}) = lims_{2}−lims_{3}.

(15) If s_{1} is convergent, then lim(z · s_{1}) = z · lims_{1}.

(16) If s_{1} is convergent, then lim(− s_{1}) = − lims_{1}.

(17) If s_{1} is convergent, then lim(s_{1} + x) = lims_{1} + x.

(18) If s_{1} is convergent, then lim(s_{1} − x) = lims_{1} − x.

(19) Suppose s_{1} is convergent. Then lims_{1} = g if and only if for every r such that r > 0 there exists m such that for every n such that n m holds ∥s_{1}(n) − g∥ < r.

Let us consider X, s_{1}. The functor ∥s_{1}∥ yielding a sequence of real numbers is defined as follows:

(Def. 3) For every n holds ∥s_{1}∥(n) = ∥s_{1}(n)∥.

One can prove the following three propositions:

(20) If s_{1} is convergent, then ∥s_{1}∥ is convergent.

(21) If s_{1} is convergent and lims_{1} = g, then ∥s_{1}∥ is convergent and lim∥s_{1}∥ = ∥g∥.

(22) If s_{1} is convergent and lims_{1} = g, then ∥s_{1} − g∥ is convergent and

lim∥s_{1} − g∥ = 0. Let us consider X, s_{1}, x. The functor ρ(s_{1}, x) yielding a sequence of real

numbers is defined as follows: (Def. 4) For every n holds (ρ(s_{1}, x))(n) = ρ(s_{1}(n), x).

One can prove the following propositions:

(23) If s_{1} is convergent and lims_{1} = g, then ρ(s_{1}, g) is convergent.

(24) If s_{1} is convergent and lims_{1} = g, then ρ(s_{1}, g) is convergent and limρ(s_{1}, g) = 0.

(25) If s_{2} is convergent and lims_{2} = g_{1} and s_{3} is convergent and lims_{3} = g_{2}, then ∥s_{2} + s_{3}∥ is convergent and lim∥s_{2} + s_{3}∥ = ∥g_{1} + g_{2}∥.

(26) If s_{2} is convergent and lims_{2} = g_{1} and s_{3} is convergent and lims_{3} = g_{2}, then ∥(s_{2} +s_{3})−(g_{1} +g_{2})∥ is convergent and lim∥(s_{2} +s_{3})−(g_{1} +g_{2})∥ = 0.

(27) If s_{2} is convergent and lims_{2} = g_{1} and s_{3} is convergent and lims_{3} = g_{2}, then ∥s_{2} − s_{3}∥ is convergent and lim∥s_{2} − s_{3}∥ = ∥g_{1} − g_{2}∥.

(28) If s_{2} is convergent and lims_{2} = g_{1} and s_{3} is convergent and lims_{3} = g_{2}, then ∥s_{2} − s_{3} − (g_{1} − g_{2})∥ is convergent and lim∥s_{2} − s_{3} − (g_{1} − g_{2})∥ = 0.

(29) If s_{1} is convergent and lims_{1} = g, then ∥z · s_{1}∥ is convergent and lim∥z · s_{1}∥ = ∥z · g∥.

(30) If s_{1} is convergent and lims_{1} = g, then ∥z · s_{1} − z · g∥ is convergent and lim∥z · s_{1} − z · g∥ = 0.

(31) If s_{1} is convergent and lims_{1} = g, then ∥− s_{1}∥ is convergent and lim∥− s_{1}∥ = ∥− g∥.

(32) If s_{1} is convergent and lims_{1} = g, then ∥− s_{1} − − g∥ is convergent and lim∥− s_{1} − − g∥ = 0.

(33) If s_{1} is convergent and lims_{1} = g, then ∥(s_{1} + x) − (g + x)∥ is convergent and lim∥(s_{1} + x) − (g + x)∥ = 0.

(34) If s_{1} is convergent and lims_{1} = g, then ∥s_{1} − x∥ is convergent and lim∥s_{1} − x∥ = ∥g − x∥.

(35) If s_{1} is convergent and lims_{1} = g, then ∥s_{1} − x − (g − x)∥ is convergent and lim∥s_{1} − x − (g − x)∥ = 0.

(36) If s_{2} is convergent and lims_{2} = g_{1} and s_{3} is convergent and lims_{3} = g_{2}, then ρ(s_{2} + s_{3}, g_{1} + g_{2}) is convergent and limρ(s_{2} + s_{3}, g_{1} + g_{2}) = 0.

(37) If s_{2} is convergent and lims_{2} = g_{1} and s_{3} is convergent and lims_{3} = g_{2}, then ρ(s_{2} − s_{3}, g_{1} − g_{2}) is convergent and limρ(s_{2} − s_{3}, g_{1} − g_{2}) = 0.

(38) If s_{1} is convergent and lims_{1} = g, then ρ(z · s_{1}, z · g) is convergent and limρ(z · s_{1}, z · g) = 0.

(39) If s_{1} is convergent and lims_{1} = g, then ρ(s_{1} + x, g + x) is convergent and

limρ(s_{1} + x, g + x) = 0. Let us consider X, x, r. The functor Ball(x, r) yields a subset of X and is

defined by: (Def. 5) Ball(x, r) = {y; y ranges over points of X: ∥x − y∥ < r}.

The functor \overline{Ball}(x, r) yielding a subset of X is defined by: (Def. 6) \overline{Ball}(x, r) = \{y; y ranges over points of X: ∥x − y∥ r\}.

The functor Sphere(x, r) yielding a subset of X is defined as follows: (Def. 7) Sphere(x, r) = {y; y ranges over points of X: ∥x − y∥ = r}.

Next we state a number of propositions:

(40) w ∈ Ball(x, r) iff ∥x − w∥ < r.

(41) w ∈ Ball(x, r) iff ρ(x, w) < r.

(42) If r > 0, then x ∈ Ball(x, r).

(43) If y ∈ Ball(x, r) and w ∈ Ball(x, r), then ρ(y, w) < 2 · r.

(44) If y ∈ Ball(x, r), then y − w ∈ Ball(x − w, r).

(45) If y ∈ Ball(x, r), then y − x ∈ Ball(0_{X}, r).

(46) If y ∈ Ball(x, r) and r q, then y ∈ Ball(x, q).

(47) w ∈ \overline{Ball}(x, r) iff ∥x − w∥ r.

(48) w ∈ \overline{Ball}(x, r) iff ρ(x, w) r.

(49) If r 0, then x ∈ \overline{Ball}(x, r).

(50) If y ∈ Ball(x, r), then y ∈ \overline{Ball}(x, r).

(51) w ∈ Sphere(x, r) iff ∥x − w∥ = r.

(52) w ∈ Sphere(x, r) iff ρ(x, w) = r.

(53) If y ∈ Sphere(x, r), then y ∈ \overline{Ball}(x, r).

(54) Ball(x, r) ⊆ \overline{Ball}(x, r).

(55) Sphere(x, r) ⊆ \overline{Ball}(x, r).

(56) Ball(x, r) ∪ Sphere(x, r) = \overline{Ball}(x, r).

2. Cauchy Sequence and Hilbert Space with Complex Coefficient

Let us consider X and let us consider s_{1}. We say that s_{1} is Cauchy if and only if:

(Def. 8) For every r such that r > 0 there exists k such that for all n, m such

that n k and m k holds ρ(s_{1}(n), s_{1}(m)) < r. The following propositions are true:

(57) If s_{1} is constant, then s_{1} is Cauchy.

(58) s_{1} is Cauchy if and only if for every r such that r > 0 there exists k such that for all n, m such that n k and m k holds ∥s_{1}(n) − s_{1}(m)∥ < r.

(59) If s_{2} is Cauchy and s_{3} is Cauchy, then s_{2} + s_{3} is Cauchy.

(60) If s_{2} is Cauchy and s_{3} is Cauchy, then s_{2} − s_{3} is Cauchy.

(61) If s_{1} is Cauchy, then z · s_{1} is Cauchy.

(62) If s_{1} is Cauchy, then − s_{1} is Cauchy.

(63) If s_{1} is Cauchy, then s_{1} + x is Cauchy.

(64) If s_{1} is Cauchy, then s_{1} − x is Cauchy.

(65) If s_{1} is convergent, then s_{1} is Cauchy.

Let us consider X and let us consider s_{2}, s_{3}. We say that s_{2} is compared to s_{3} if and only if:

(Def. 9) For every r such that r > 0 there exists m such that for every n such

that n m holds ρ(s_{2}(n), s_{3}(n)) < r. One can prove the following two propositions:

(66) s_{1} is compared to s_{1}.

(67) If s_{2} is compared to s_{3}, then s_{3} is compared to s_{2}.

Let us consider X and let us consider s_{2}, s_{3}. Let us notice that the predicate s_{2} is compared to s_{3} is reflexive and symmetric.

The following propositions are true:

(68) If s_{2} is compared to s_{3} and s_{3} is compared to s_{4}, then s_{2} is compared to s_{4}.

(69) s_{2} is compared to s_{3} iff for every r such that r > 0 there exists m such that for every n such that n m holds ∥s_{2}(n) − s_{3}(n)∥ < r.

(70) If there exists k such that for every n such that n k holds s_{2}(n) = s_{3}(n), then s_{2} is compared to s_{3}.

(71) If s_{2} is Cauchy and compared to s_{3}, then s_{3} is Cauchy.

(72) If s_{2} is convergent and compared to s_{3}, then s_{3} is convergent.

(73) If s_{2} is convergent and lims_{2} = g and s_{2} is compared to s_{3}, then s_{3} is

convergent and lims_{3} = g. Let us consider X and let us consider s_{1}. We say that s_{1} is bounded if and

only if: (Def. 10) There exists M such that M > 0 and for every n holds ∥s_{1}(n)∥ M.

We now state several propositions:

(74) If s_{2} is bounded and s_{3} is bounded, then s_{2} + s_{3} is bounded.

(75) If s_{1} is bounded, then − s_{1} is bounded.

(76) If s_{2} is bounded and s_{3} is bounded, then s_{2} − s_{3} is bounded.

(77) If s_{1} is bounded, then z · s_{1} is bounded.

(78) If s_{1} is constant, then s_{1} is bounded.

(79) For every m there exists M such that M > 0 and for every n such that n m holds ∥s_{1}(n)∥ < M.

(80) If s_{1} is convergent, then s_{1} is bounded.

(81) If s_{2} is bounded and compared to s_{3}, then s_{3} is bounded.

Let us consider X, N_{1}, s_{1}. Then s_{1} · N_{1} is a sequence of X.

We now state several propositions:

(82) Let X be a complex unitary space, s be a sequence of X, N be an increasing sequence of naturals, and n be a natural number. Then (s · N)(n) = s(N(n)).

(83) s_{1} is a subsequence of s_{1}.

(84) If s_{2} is a subsequence of s_{3} and s_{3} is a subsequence of s_{4}, then s_{2} is a subsequence of s_{4}.

(85) If s_{1} is constant and s_{2} is a subsequence of s_{1}, then s_{2} is constant.

(86) If s_{1} is constant and s_{2} is a subsequence of s_{1}, then s_{1} = s_{2}.

(87) If s_{1} is bounded and s_{2} is a subsequence of s_{1}, then s_{2} is bounded.

(88) If s_{1} is convergent and s_{2} is a subsequence of s_{1}, then s_{2} is convergent.

(89) If s_{2} is a subsequence of s_{1} and s_{1} is convergent, then lims_{2} = lims_{1}.

(90) If s_{1} is Cauchy and s_{2} is a subsequence of s_{1}, then s_{2} is Cauchy. Let us consider X, let us consider s_{1}, and let us consider k. The functor

s_{1} ↑ k yields a sequence of X and is defined as follows: (Def. 11) For every n holds (s_{1} ↑ k)(n) = s_{1}(n + k).

One can prove the following propositions:

(91) s_{1} ↑ 0 = s_{1}.

(92) s_{1} ↑ k ↑ m = s_{1} ↑ m ↑ k.

(93) s_{1} ↑ k ↑ m = s_{1} ↑ (k + m).

(94) (s_{2} + s_{3}) ↑ k = s_{2} ↑ k + s_{3} ↑ k.

(95) (− s_{1}) ↑ k = − s_{1} ↑ k.

(96) (s_{2} − s_{3}) ↑ k = s_{2} ↑ k − s_{3} ↑ k.

(97) (z · s_{1}) ↑ k = z · (s_{1} ↑ k).

(98) (s_{1} · N_{1}) ↑ k = s_{1} · (N_{1} ↑ k).

(99) s_{1} ↑ k is a subsequence of s_{1}.

(100) If s_{1} is convergent, then s_{1} ↑ k is convergent and lim(s_{1} ↑ k) = lims_{1}.

(101) If s_{1} is convergent and there exists k such that s_{1} = s_{2} ↑ k, then s_{2} is convergent.

(102) If s_{1} is Cauchy and there exists k such that s_{1} = s_{2}↑k, then s_{2} is Cauchy.

(103) If s_{1} is Cauchy, then s_{1} ↑ k is Cauchy.

(104) If s_{2} is compared to s_{3}, then s_{2} ↑ k is compared to s_{3} ↑ k.

(105) If s_{1} is bounded, then s_{1} ↑ k is bounded.

(106) If s_{1} is constant, then s_{1} ↑ k is constant.

Let us consider X. We say that X is complete if and only if: (Def. 12) For every s_{1} such that s_{1} is Cauchy holds s_{1} is convergent.

The following proposition is true

(107) If X is complete and s_{1} is Cauchy, then s_{1} is bounded.

Let us consider X. We say that X is Hilbert if and only if: (Def. 13) X is a complex unitary space and complete. convergent sequences in complex unitary space 165

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Recursive Definitions. Part II^{1}

Artur Korniłowicz University of Białystok

MML Identifier: RECDEF 2.

The papers [7], [4], [9], [8], [5], [6], [1], [10], [2], [11], and [3] provide the terminology and notation for this paper.

In this paper a, b, c, d, e, z, A, B, C, D, E are sets.

Let x be a set. Let us assume that there exist sets x_{1}, x_{2}, x_{3} such that x = ⟨⟨x_{1}, x_{2}, x_{3}⟩⟩. The functor x_{1,3} is defined as follows:

(Def. 1) For all sets y_{1}, y_{2}, y_{3} such that x = ⟨⟨y_{1}, y_{2}, y_{3}⟩⟩ holds x_{1,3} = y_{1}.

The functor x_{2,3} is defined by: (Def. 2) For all sets y_{1}, y_{2}, y_{3} such that x = ⟨⟨y_{1}, y_{2}, y_{3}⟩⟩ holds x_{2,3} = y_{2}.

The functor x_{3,3} is defined by: (Def. 3) For all sets y_{1}, y_{2}, y_{3} such that x = ⟨⟨y_{1}, y_{2}, y_{3}⟩⟩ holds x_{3,3} = y_{3}.

The following propositions are true:

(1) If there exist a, b, c such that z = ⟨⟨a, b, c⟩⟩, then z = ⟨⟨z_{1,3}, z_{2,3}, z_{3,3} ⟩⟩.

(2) If z ∈ [: A, B, C:], then z_{1,3} ∈ A and z_{2,3} ∈ B and z_{3,3} ∈ C.

(3) If z ∈ [: A, B, C:], then z = ⟨⟨z_{1,3}, z_{2,3}, z_{3,3} ⟩⟩.

Let x be a set. Let us assume that there exist sets x_{1}, x_{2}, x_{3}, x_{4} such that x = ⟨⟨x_{1}, x_{2}, x_{3}, x_{4}⟩⟩. The functor x_{1,4} is defined by:

(Def. 4) For all sets y_{1}, y_{2}, y_{3}, y_{4} such that x = ⟨⟨y_{1}, y_{2}, y_{3}, y_{4}⟩⟩ holds x_{1,4} = y_{1}.

The functor x_{2,4} is defined by: (Def. 5) For all sets y_{1}, y_{2}, y_{3}, y_{4} such that x = ⟨⟨y_{1}, y_{2}, y_{3}, y_{4}⟩⟩ holds x_{2,4} = y_{2}.

The functor x_{3,4} is defined as follows: (Def. 6) For all sets y_{1}, y_{2}, y_{3}, y_{4} such that x = ⟨⟨y_{1}, y_{2}, y_{3}, y_{4}⟩⟩ holds x_{3,4} = y_{3}.

The functor x_{4,4} is defined as follows:

1The paper was written during author’s post-doctoral fellowship granted by Shinshu University, Japan.

(Def. 7) For all sets y_{1}, y_{2}, y_{3}, y_{4} such that x = ⟨⟨y_{1}, y_{2}, y_{3}, y_{4}⟩⟩ holds x_{4,4} = y_{4}. Next we state three propositions:

(4) If there exist a, b, c, d such that z = ⟨⟨a, b, c, d⟩⟩, then z = ⟨⟨z_{1,4}, z_{2,4}, z_{3,4}, z_{4,4} ⟩⟩.

(5) If z ∈ [: A, B, C, D:], then z_{1,4} ∈ A and z_{2,4} ∈ B and z_{3,4} ∈ C and z_{4,4} ∈ D.

(6) If z ∈ [: A, B, C, D:], then z = ⟨⟨z_{1,4}, z_{2,4}, z_{3,4}, z_{4,4} ⟩⟩.

Let x be a set. Let us assume that there exist sets x_{1}, x_{2}, x_{3}, x_{4}, x_{5} such that x = ⟨⟨x_{1}, x_{2}, x_{3}, x_{4}, x_{5}⟩⟩. The functor x_{1,5} is defined by:

(Def. 8) For all sets y_{1}, y_{2}, y_{3}, y_{4}, y_{5} such that x = ⟨⟨y_{1}, y_{2}, y_{3}, y_{4}, y_{5}⟩⟩holds x_{1,5} =

y_{1}. The functor x_{2,5} is defined by:

(Def. 9) For all sets y_{1}, y_{2}, y_{3}, y_{4}, y_{5} such that x = ⟨⟨y_{1}, y_{2}, y_{3}, y_{4}, y_{5}⟩⟩holds x_{2,5} =

y_{2}. The functor x_{3,5} is defined as follows:

(Def. 10) For all sets y_{1}, y_{2}, y_{3}, y_{4}, y_{5} such that x = ⟨⟨y_{1}, y_{2}, y_{3}, y_{4}, y_{5}⟩⟩holds x_{3,5} =

y_{3}. The functor x_{4,5} is defined as follows:

(Def. 11) For all sets y_{1}, y_{2}, y_{3}, y_{4}, y_{5} such that x = ⟨⟨y_{1}, y_{2}, y_{3}, y_{4}, y_{5}⟩⟩holds x_{4,5} =

y_{4}. The functor x_{5,5} is defined by:

(Def. 12) For all sets y_{1}, y_{2}, y_{3}, y_{4}, y_{5} such that x = ⟨⟨y_{1}, y_{2}, y_{3}, y_{4}, y_{5}⟩⟩holds x_{5,5} =

y_{5}. The following propositions are true:

(7) If there exist a, b, c, d, e such that z = ⟨⟨a, b, c, d, e⟩⟩, then z = ⟨⟨z_{1,5}, z_{2,5}, z_{3,5}, z_{4,5}, z_{5,5} ⟩⟩.

(8) If z ∈ [: A, B, C, D, E:], then z_{1,5} ∈ A and z_{2,5} ∈ B and z_{3,5} ∈ C and z_{4,5} ∈ D and z_{5,5} ∈ E.

(9) If z ∈ [: A, B, C, D, E:], then z = ⟨⟨z_{1,5}, z_{2,5}, z_{3,5}, z_{4,5}, z_{5,5} ⟩⟩.

In this article we present several logical schemes. The scheme ExFunc3Cond deals with a set A, three unary functors F, G, and H yielding sets, and three unary predicates P, Q, R, and states that:

There exists a function f such that domf = A and for every set c such that c ∈ A holds if P[c], then f(c) = F(c) and if Q[c], then f(c) = G(c) and if R[c], then f(c) = H(c)

provided the parameters meet the following conditions:

• For every set c such that c ∈ A holds if P[c], then not Q[c] and if P[c], then not R[c] and if Q[c], then not R[c], and

• For every set c such that c ∈ A holds P[c] or Q[c] or R[c].

The scheme ExFunc4Cond deals with a set A, four unary functors F, G, H, and I yielding sets, and four unary predicates P, Q, R, S, and states that:

There exists a function f such that

(i) domf = A, and (ii) for every set c such that c ∈ A holds if P[c], then f(c) = F(c) and if Q[c], then f(c) = G(c) and if R[c], then f(c) = H(c) and if S[c], then f(c) = I(c)

provided the following conditions are satisfied:

• Let c be a set such that c ∈ A. Then (i) if P[c], then not Q[c], (ii) if P[c], then not R[c], (iii) if P[c], then not S[c], (iv) if Q[c], then not R[c],

(v) if Q[c], then not S[c], and (vi) if R[c], then not S[c],

and

• For every set c such that c ∈ A holds P[c] or Q[c] or R[c] or S[c]. The scheme DoubleChoiceRec deals with non empty sets A, B, an element

C of A, an element D of B, and a 5-ary predicate P, and states that: There exists a function f from ℕ into A and there exists a function g from ℕ into B such that f(0) = C and g(0) = D and for every element n of ℕ holds P[n, f(n), g(n), f(n + 1), g(n + 1)]

provided the parameters satisfy the following condition:

• Let n be an element of ℕ, x be an element of A, and y be an element of B. Then there exists an element x_{1} of A and there exists an element y_{1} of B such that P[n, x, y, x_{1}, y_{1}].

The scheme LambdaRec2Ex deals with sets A, B and a ternary functor F yielding a set, and states that:

There exists a function f such that domf = ℕ and f(0) = A and f(1) = B and for every natural number n holds f(n + 2) = F(n, f(n), f(n + 1))

for all values of the parameters.

The scheme LambdaRec2ExD deals with a non empty set A, elements B, C of A, and a ternary functor F yielding an element of A, and states that:

There exists a function f from ℕ into A such that f(0) = B and f(1) = C and for every natural number n holds f(n + 2) = F(n, f(n), f(n + 1))

for all values of the parameters.

The scheme LambdaRec2Un deals with sets A, B, functions C, D, and a ternary functor C yielding a set, and states that:

C = D provided the parameters meet the following requirements:

• domC = ℕ,

• C(0) = A and C(1) = B,

• For every natural number n holds C(n + 2) = C(n, C(n), C(n + 1)),

• domD = ℕ,

• D(0) = A and D(1) = B, and

• For every natural number n holds D(n+2) = C(n, D(n), D(n+1)). The scheme LambdaRec2UnD deals with a non empty set A, elements B, C

of A, functions D, E from ℕ into A, and a ternary functor D yielding an element of A, and states that:

D = E provided the following requirements are met:

• D(0) = B and D(1) = C,

• For every natural number n holds D(n+2) = D(n, D(n), D(n+1)),

• E(0) = B and E(1) = C, and

• For every natural number n holds E(n+2) = D(n, E(n), E(n+1)). The scheme LambdaRec3Ex deals with sets A, B, C and a 4-ary functor F

yielding a set, and states that:

There exists a function f such that domf = ℕ and f(0) = A and f(1) = B and f(2) = C and for every natural number n holds f(n + 3) = F(n, f(n), f(n + 1), f(n + 2))

for all values of the parameters.

The scheme LambdaRec3ExD deals with a non empty set A, elements B, C, D of A, and a 4-ary functor F yielding an element of A, and states that:

There exists a function f from ℕ into A such that f(0) = B and f(1) = C and f(2) = D and for every natural number n holds f(n + 3) = F(n, f(n), f(n + 1), f(n + 2))

for all values of the parameters.

The scheme LambdaRec3Un deals with sets A, B, C, functions D, E, and a 4-ary functor D yielding a set, and states that:

D = E provided the parameters meet the following requirements:

• domD = ℕ,

• D(0) = A and D(1) = B and D(2) = C,

• For every natural number n holds D(n + 3) = D(n, D(n), D(n + 1), D(n + 2)),

• domE = ℕ,

• E(0) = A and E(1) = B and E(2) = C, and

• For every natural number n holds E(n + 3) = D(n, E(n), E(n + 1), E(n + 2)).

The scheme LambdaRec3UnD deals with a non empty set A, elements B, C, D of A, functions E, F from ℕ into A, and a 4-ary functor E yielding an element of A, and states that:

E = F provided the parameters meet the following requirements:

• E(0) = B and E(1) = C and E(2) = D,

• For every natural number n holds E(n + 3) = E(n, E(n), E(n + 1), E(n + 2)),

• F(0) = B and F(1) = C and F(2) = D, and

• For every natural number n holds F(n + 3) = E(n, F(n), F(n + 1), F(n + 2)).

The scheme LambdaRec4Ex deals with sets A, B, C, D and a 5-ary functor F yielding a set, and states that:

There exists a function f such that domf = ℕ and f(0) = A and f(1) = B and f(2) = C and f(3) = D and for every natural number n holds f(n+4) = F(n, f(n), f(n+1), f(n+2), f(n+3))

for all values of the parameters.

The scheme LambdaRec4ExD deals with a non empty set A, elements B, C, D, E of A, and a 5-ary functor F yielding an element of A, and states that:

There exists a function f from ℕ into A such that f(0) = B and f(1) = C and f(2) = D and f(3) = E and for every natural number n holds f(n+4) = F(n, f(n), f(n+1), f(n+2), f(n+3))

for all values of the parameters.

The scheme LambdaRec4Un deals with sets A, B, C, D, functions E, F, and a 5-ary functor E yielding a set, and states that:

E = F provided the parameters satisfy the following conditions:

• domE = ℕ,

• E(0) = A and E(1) = B and E(2) = C and E(3) = D,

• For every natural number n holds E(n + 4) = E(n, E(n), E(n + 1), E(n + 2), E(n + 3)),

• domF = ℕ,

• F(0) = A and F(1) = B and F(2) = C and F(3) = D, and

• For every natural number n holds F(n + 4) = E(n, F(n), F(n + 1), F(n + 2), F(n + 3)).

The scheme LambdaRec4UnD deals with a non empty set A, elements B, C, D, E of A, functions F, G from ℕ into A, and a 5-ary functor F yielding an element of A, and states that:

F = G provided the parameters meet the following requirements:

• F(0) = B and F(1) = C and F(2) = D and F(3) = E,

• For every natural number n holds F(n + 4) = F(n, F(n), F(n + 1), F(n + 2), F(n + 3)),

• G(0) = B and G(1) = C and G(2) = D and G(3) = E, and 172 artur korniłowicz

• For every natural number n holds G(n + 4) = F(n, G(n), G(n + 1), G(n + 2), G(n + 3)).

References

[1] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41–46, 1990.

[2] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55– 65, 1990.

[3] Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153–164, 1990.

[4] Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47–53, 1990.

[5] Michał Muzalewski and Wojciech Skaba. N-tuples and Cartesian products for n=5. Formalized Mathematics, 2(2):255–258, 1991.

[6] Andrzej Trybulec. Subsets of complex numbers. To appear in Formalized Mathematics.

[7] Andrzej Trybulec. Tarski Grothendieck set theory. Formalized Mathematics, 1(1):9–11, 1990.

[8] Andrzej Trybulec. Tuples, projections and Cartesian products. Formalized Mathematics, 1(1):97–105, 1990.

[9] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67–71, 1990.

[10] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73–83, 1990.

[11] Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181–186, 1990.

The Exponential Function on Banach Algebra

Yasunari Shidama Shinshu University Nagano

Summary. In this article, the basic properties of the exponential function on Banach algebra are described.

MML Identifier: LOPBAN 4.

The notation and terminology used here are introduced in the following papers: [17], [19], [20], [3], [4], [2], [16], [5], [1], [18], [9], [11], [12], [8], [6], [7], [13], [10], [21], [14], and [15].

For simplicity, we use the following convention: X denotes a Banach algebra, p denotes a real number, w, z, z_{1}, z_{2} denote elements of X, k, l, m, n denote natural numbers, s_{1}, s_{2}, s_{3}, s, s′ denote sequences of X, and r_{1} denotes a sequence of real numbers.

Let X be a non empty normed algebra structure and let x, y be elements of X. We say that x, y are commutative if and only if:

(Def. 1) x · y = y · x.

Let us note that the predicate x, y are commutative is symmetric.

Next we state a number of propositions:

(1) If s_{2} is convergent and s_{3} is convergent and lim(s_{2} − s_{3}) = 0_{X}, then lims_{2} = lims_{3}.

(2) For every z such that for every natural number n holds s(n) = z holds lims = z.

(3) If s is convergent and s′ is convergent, then s · s′ is convergent.

(4) If s is convergent, then z · s is convergent.

(5) If s is convergent, then s · z is convergent.

(6) If s is convergent, then lim(z · s) = z · lims.

(7) If s is convergent, then lim(s · z) = lims · z.

(8) If s is convergent and s′ is convergent, then lim(s · s′) = lims · lims′.

(9) (∑_{α=0}^{κ}(z·s_{1})(α))_{κ∈ℕ} = z·(∑_{α=0}^{κ}(s_{1})(α))_{κ∈ℕ} and (∑_{α=0}^{κ}(s_{1}·z)(α))_{κ∈ℕ} = (∑_{α=0}^{κ}(s_{1})(α))_{κ∈ℕ} · z.

(10) ∥(∑_{α=0}^{κ}(s_{1})(α))_{κ∈ℕ}(k)∥ (∑_{α=0}^{κ}∥s_{1}∥(α))_{κ∈ℕ}(k).

(11) If for every n such that n m holds s_{2}(n) = s_{3}(n), then (∑_{α=0}^{κ}(s_{2})(α))_{κ∈ℕ}(m) = (∑_{α=0}^{κ}(s_{3})(α))_{κ∈ℕ}(m).

(12) If for every n holds ∥s_{1}(n)∥ r_{1}(n) and r_{1} is convergent and limr_{1} = 0,

then s_{1} is convergent and lims_{1} = 0_{X}. Let us consider X and let z be an element of X. The functor z ExpSeq

yielding a sequence of X is defined as follows: (Def. 2) For every n holds z ExpSeq(n) = \frac{1}{n!} · z_{ℕ}^{n}.

The scheme ExNormSpace CASE deals with a non empty Banach algebra A and a binary functor F yielding a point of A, and states that:

For every k there exists a sequence s_{1} of A such that for every n holds if n k, then s_{1}(n) = F(k, n) and if n > k, then s_{1}(n) = 0_{A}

for all values of the parameters.

Next we state the proposition

(13) For every k such that 0 < k holds (k − ′1)! · k = k! and for all m, k such

that k m holds (m − ′k)! · ((m + 1) − k) = ((m + 1) − ′k)!. Let n be a natural number. The functor Coef n yields a sequence of real

numbers and is defined by: (Def. 3) For every natural number k holds if k n, then (Coef n)(k) = \frac{n!}{k!·(n− ′k)!}

and if k > n, then (Coef n)(k) = 0. Let n be a natural number. The functor Coef en yielding a sequence of real

numbers is defined by: (Def. 4) For every natural number k holds if k n, then (Coef en)(k) = \frac{1}{k!·(n− ′k)!}

and if k > n, then (Coef en)(k) = 0. Let us consider X, s_{1}. The functor Shifts_{1} yielding a sequence of X is defined

as follows: (Def. 5) (Shifts_{1})(0) = 0_{X} and for every natural number k holds (Shifts_{1})(k +

1) = s_{1}(k). Let us consider n, let us consider X, and let z, w be elements of X. The

functor Expan(n, z, w) yields a sequence of X and is defined by: (Def. 6) For every natural number k holds if k n, then (Expan(n, z, w))(k) =

(Coef n)(k) · z_{ℕ}^{k} · w_{ℕ}^{n− ′k} and if n < k, then (Expan(n, z, w))(k) = 0_{X}. Let us consider n, let us consider X, and let z, w be elements of X. The

functor Expan e(n, z, w) yields a sequence of X and is defined as follows:

(Def. 7) For every natural number k holds if k n, then (Expan e(n, z, w))(k) =

(Coef en)(k) · z_{ℕ}^{k} · w_{ℕ}^{n− ′k} and if n < k, then (Expan e(n, z, w))(k) = 0_{X}. Let us consider n, let us consider X, and let z, w be elements of X. The

functor Alfa(n, z, w) yields a sequence of X and is defined as follows: (Def. 8) For every natural number k holds if k n, then (Alfa(n, z, w))(k) =

z ExpSeq(k) · (∑_{α=0}^{κ} w ExpSeq(α))_{κ∈ℕ}(n − ′ k) and if n < k, then (Alfa(n, z, w))(k) = 0_{X}.

Let us consider X, let z, w be elements of X, and let n be a natural number. The functor Conj(n, z, w) yields a sequence of X and is defined by:

(Def. 9) For every natural number k holds if k n, then (Conj(n, z, w))(k) = z ExpSeq(k)·((∑_{α=0}^{κ} w ExpSeq(α))_{κ∈ℕ}(n)−(∑_{α=0}^{κ} w ExpSeq(α))_{κ∈ℕ}(n− ′ k)) and if n < k, then (Conj(n, z, w))(k) = 0_{X}.

One can prove the following propositions:

(14) z ExpSeq(n + 1) = \frac{1}{n+1} · z · z ExpSeq(n) and z ExpSeq(0) = 1_{X} and ∥z ExpSeq(n)∥ ∥z∥ ExpSeq(n).

(15) If 0 < k, then (Shifts_{1})(k) = s_{1}(k − ′1).

(16) (∑_{α=0}^{κ}(s_{1})(α))_{κ∈ℕ}(k) = (∑_{α=0}^{κ}(Shifts_{1})(α))_{κ∈ℕ}(k) + s_{1}(k).

(17) For all z, w such that z, w are commutative holds (z + w)_{ℕ}^{n} = (∑_{α=0}^{κ}(Expan(n, z, w))(α))_{κ∈ℕ}(n).

(18) Expan e(n, z, w) = \frac{1}{n!} · Expan(n, z, w).

(19) For all z, w such that z, w are commutative holds \frac{1}{n!} · (z + w)_{ℕ}^{n} = (∑_{α=0}^{κ}(Expan e(n, z, w))(α))_{κ∈ℕ}(n).

(20) 0_{X} ExpSeq is norm-summable and ∑(0_{X} ExpSeq) = 1_{X}.

Let us consider X and let z be an element of X. Observe that z ExpSeq is norm-summable.

Next we state a number of propositions:

(21) z ExpSeq(0) = 1_{X} and (Expan(0, z, w))(0) = 1_{X}.

(22) If l k, then (Alfa(k + 1, z, w))(l) = (Alfa(k, z, w))(l) + (Expan e(k + 1, z, w))(l).

(23) (∑_{α=0}^{κ}(Alfa(k + 1, z, w))(α))_{κ∈ℕ}(k) = (∑_{α=0}^{κ}(Alfa(k, z, w))(α))_{κ∈ℕ}(k) + (∑_{α=0}^{κ}(Expan e(k + 1, z, w))(α))_{κ∈ℕ}(k).

(24) z ExpSeq(k) = (Expan e(k, z, w))(k).

(25) For all z, w such that z, w are commutative holds (∑_{α=0}^{κ} z + w ExpSeq(α))_{κ∈ℕ}(n) = (∑_{α=0}^{κ}(Alfa(n, z, w))(α))_{κ∈ℕ}(n).

(26) For all z, w such that z, w are commutative holds (∑_{α=0}^{κ} z ExpSeq(α))_{κ∈ℕ}(k) · (∑_{α=0}^{κ} w ExpSeq(α))_{κ∈ℕ}(k) − (∑_{α=0}^{κ} z + w ExpSeq(α))_{κ∈ℕ}(k) = (∑_{α=0}^{κ}(Conj(k, z, w))(α))_{κ∈ℕ}(k).

(27) 0 ∥z∥ ExpSeq(n).

(28) ∥(∑_{α=0}^{κ} z ExpSeq(α))_{κ∈ℕ}(k)∥ (∑_{α=0}^{κ}∥z∥ ExpSeq(α))_{κ∈ℕ}(k) and (∑_{α=0}^{κ}∥z∥ ExpSeq(α))_{κ∈ℕ}(k) ∑(∥z∥ ExpSeq) and ∥(∑_{α=0}^{κ} z ExpSeq(α))_{κ∈ℕ}(k)∥ ∑(∥z∥ ExpSeq).

(29) 1 ∑(∥z∥ ExpSeq).

(30) |(∑_{α=0}^{κ}∥z∥ ExpSeq(α))_{κ∈ℕ}(n)| = (∑_{α=0}^{κ}∥z∥ ExpSeq(α))_{κ∈ℕ}(n) and if n m, then |(∑_{α=0}^{κ}∥z∥ ExpSeq(α))_{κ∈ℕ}(m)−(∑_{α=0}^{κ}∥z∥ ExpSeq(α))_{κ∈ℕ}(n)| = (∑_{α=0}^{κ}∥z∥ ExpSeq(α))_{κ∈ℕ}(m) − (∑_{α=0}^{κ}∥z∥ ExpSeq(α))_{κ∈ℕ}(n).

(31) |(∑_{α=0}^{κ}∥Conj(k, z, w)∥(α))_{κ∈ℕ}(n)| = (∑_{α=0}^{κ}∥Conj(k, z, w)∥(α))_{κ∈ℕ}(n).

(32) For every real number p such that p > 0 there exists n such that for every k such that n k holds |(∑_{α=0}^{κ}∥Conj(k, z, w)∥(α))_{κ∈ℕ}(k)| < p.

(33) For every s_{1} such that for every k holds s_{1}(k) =

(∑_{α=0}^{κ}(Conj(k, z, w))(α))_{κ∈ℕ}(k) holds s_{1} is convergent and lims_{1} = 0_{X}. Let X be a Banach algebra. The functor expX yielding a function from the

carrier of X into the carrier of X is defined by: (Def. 10) For every element z of the carrier of X holds (expX)(z) = ∑(z ExpSeq).

Let us consider X, z. The functor expz yields an element of X and is defined by:

(Def. 11) expz = (expX)(z).

One can prove the following propositions:

(34) For every z holds expz = ∑(z ExpSeq).

(35) Let given z_{1}, z_{2}. Suppose z_{1}, z_{2} are commutative. Then exp(z_{1} + z_{2}) = expz_{1} ·expz_{2} and exp(z_{2} +z_{1}) = expz_{2} ·expz_{1} and exp(z_{1} +z_{2}) = exp(z_{2} + z_{1}) and expz_{1}, expz_{2} are commutative.

(36) For all z_{1}, z_{2} such that z_{1}, z_{2} are commutative holds z_{1}·expz_{2} = expz_{2}·z_{1}.

(37) exp(0_{X}) = 1_{X}.

(38) expz · exp(− z) = 1_{X} and exp(− z) · expz = 1_{X}.

(39) expz is invertible and (expz)^{−1} = exp(− z) and exp(− z) is invertible and (exp(− z))^{−1} = expz.

(40) For every z and for all real numbers s, t holds s·z, t·z are commutative.

(41) Let given z and s, t be real numbers. Then exp(s · z) · exp(t · z) = exp((s+t)·z) and exp(t·z)·exp(s·z) = exp((t+s)·z) and exp((s+t)·z) = exp((t + s) · z) and exp(s · z), exp(t · z) are commutative.

References

[1] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41–46, 1990.

[2] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91–96, 1990.

[3] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55– 65, 1990.

[4] Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153–164,

the exponential function on banach algebra 177

[5] Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35–40, 1990.

[6] Jarosław Kotowicz. Convergent sequences and the limit of sequences. Formalized Mathematics, 1(2):273–275, 1990.

[7] Jarosław Kotowicz. Monotone real sequences. Subsequences. Formalized Mathematics, 1(3):471–475, 1990.

[8] Jarosław Kotowicz. Real sequences and basic operations on them. Formalized Mathematics, 1(2):269–272, 1990.

[9] Eugeniusz Kusak, Wojciech Leończuk, and Michał Muzalewski. Abelian groups, fields and vector spaces. Formalized Mathematics, 1(2):335–342, 1990.

[10] Takaya Nishiyama and Yasuho Mizuhara. Binary arithmetics. Formalized Mathematics, 4(1):83–86, 1993.

[11] Jan Popiołek. Some properties of functions modul and signum. Formalized Mathematics, 1(2):263–264, 1990.

[12] Jan Popiołek. Real normed space. Formalized Mathematics, 2(1):111–115, 1991.

[13] Konrad Raczkowski and Andrzej Nędzusiak. Series. Formalized Mathematics, 2(4):449– 452, 1991.

[14] Yasunari Shidama. The Banach algebra of bounded linear operators. Formalized Mathematics, 12(2):103–108, 2004.

[15] Yasunari Shidama. The series on Banach algebra. Formalized Mathematics, 12(2):131– 138, 2004.

[16] Andrzej Trybulec. Subsets of complex numbers. To appear in Formalized Mathematics.

[17] Andrzej Trybulec. Tarski Grothendieck set theory. Formalized Mathematics, 1(1):9–11, 1990.

[18] Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291–296, 1990.

[19] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67–71, 1990.

[20] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73–83, 1990.

[21] Yuguang Yang and Yasunari Shidama. Trigonometric functions and existence of circle ratio. Formalized Mathematics, 7(2):255–263, 1998.

Fundamental Theorem of Arithmetic^{1}

Artur Korniłowicz University of Białystok

Piotr Rudnicki University of Alberta Edmonton

Summary. We formalize the notion of the prime-power factorization of a natural number and prove the Fundamental Theorem of Arithmetic. We prove also how prime-power factorization can be used to compute: products, quotients, powers, greatest common divisors and least common multiples.

MML Identifier: NAT 3.

The notation and terminology used in this paper are introduced in the following papers: [25], [27], [12], [7], [3], [4], [1], [24], [13], [2], [19], [18], [28], [8], [9], [6], [16], [15], [11], [26], [22], [23], [10], [14], [20], [5], [21], and [17].

1. Preliminaries

We follow the rules: a, b, n denote natural numbers, r denotes a real number, and f denotes a finite sequence of elements of ℝ.

Let X be an empty set. Observe that cardX is empty. One can check that every binary relation which is natural-yielding is also

real-yielding.

Let us mention that there exists a finite sequence which is natural-yielding. Let a be a non empty natural number and let b be a natural number. Observe

that a^{b} is non empty.

One can verify that every prime number is non empty. In the sequel p denotes a prime number. One can verify that Prime is infinite. The following propositions are true:

1A. Korniłowicz has been supported by a post-doctoral fellowship at Shinshu University, Nagano, Japan. P. Rudnicki has been supported by NSERC Grant OGP9207.

(1) For all natural numbers a, b, c, d such that a | c and b | d holds a·b | c·d.

(2) If 1 < a, then b a^{b}.

(3) If a ≠ 0, then n | n^{a}.

(4) For all natural numbers i, j, m, n such that i < j and m^{j} | n holds m^{i+1} | n.

(5) If p | a^{b}, then p | a.

(6) For every prime number a such that a | p^{b} holds a = p.

(7) For every finite sequence f of elements of ℕ such that a ∈ rngf holds a | ∏ f.

(8) For every finite sequence f of elements of Prime such that p | ∏ f holds p ∈ rngf.

Let f be a real-yielding finite sequence and let a be a natural number. The functor f^{a} yielding a finite sequence is defined as follows:

(Def. 1) len(f^{a}) = lenf and for every set i such that i ∈ dom(f^{a}) holds f^{a}(i) =

f(i)^{a}. Let f be a real-yielding finite sequence and let a be a natural number. One

can verify that f^{a} is real-yielding.

Let f be a natural-yielding finite sequence and let a be a natural number. Note that f^{a} is natural-yielding.

Let f be a finite sequence of elements of ℝ and let a be a natural number. Then f^{a} is a finite sequence of elements of ℝ.

Let f be a finite sequence of elements of ℕ and let a be a natural number. Then f^{a} is a finite sequence of elements of ℕ.

Next we state several propositions:

(9) f^{0} = lenf ↦ 1.

(10) f^{1} = f.

(11) (ε_{ℝ})^{a} = ε_{ℝ}.

(12) ⟨r⟩^{a} = ⟨r^{a}⟩.

(13) (f ^{⌢} ⟨r⟩)^{a} = (f^{a})^{⌢} ⟨r⟩^{a}.

(14) ∏(f^{b+1}) = ∏(f^{b}) · ∏ f.

(15) ∏(f^{a}) = (∏ f)^{a}.

2. More about Bags

Let X be a set. Note that there exists a many sorted set indexed by X which is natural-yielding and finite-support.

Let X be a set, let b be a real-yielding many sorted set indexed by X, and let a be a natural number. The functor a · b yielding a many sorted set indexed by X is defined as follows:

(Def. 2) For every set i holds (a · b)(i) = a · b(i).

Let X be a set, let b be a real-yielding many sorted set indexed by X, and let a be a natural number. One can verify that a · b is real-yielding.

Let X be a set, let b be a natural-yielding many sorted set indexed by X, and let a be a natural number. Note that a · b is natural-yielding.

Let X be a set and let b be a real-yielding many sorted set indexed by X. Note that support(0 · b) is empty.

Next we state the proposition

(16) For every set X and for every real-yielding many sorted set b indexed by

X such that a ≠ 0 holds supportb = support(a · b). Let X be a set, let b be a real-yielding finite-support many sorted set indexed

by X, and let a be a natural number. One can check that a · b is finite-support.

Let X be a set and let b_{1}, b_{2} be real-yielding many sorted sets indexed by X. The functor min(b_{1}, b_{2}) yields a many sorted set indexed by X and is defined by:

(Def. 3) For every set i holds if b_{1}(i) b_{2}(i), then (min(b_{1}, b_{2}))(i) = b_{1}(i) and if

b_{1}(i) > b_{2}(i), then (min(b_{1}, b_{2}))(i) = b_{2}(i). Let X be a set and let b_{1}, b_{2} be real-yielding many sorted sets indexed by

X. Note that min(b_{1}, b_{2}) is real-yielding.

Let X be a set and let b_{1}, b_{2} be natural-yielding many sorted sets indexed by X. Observe that min(b_{1}, b_{2}) is natural-yielding.

We now state the proposition

(17) For every set X and for all real-yielding finite-support many sorted sets

b_{1}, b_{2} indexed by X holds supportmin(b_{1}, b_{2}) ⊆ supportb_{1} ∪ supportb_{2}. Let X be a set and let b_{1}, b_{2} be real-yielding finite-support many sorted sets

indexed by X. Observe that min(b_{1}, b_{2}) is finite-support.

Let X be a set and let b_{1}, b_{2} be real-yielding many sorted sets indexed by X. The functor max(b_{1}, b_{2}) yielding a many sorted set indexed by X is defined as follows:

(Def. 4) For every set i holds if b_{1}(i) b_{2}(i), then (max(b_{1}, b_{2}))(i) = b_{2}(i) and if

b_{1}(i) > b_{2}(i), then (max(b_{1}, b_{2}))(i) = b_{1}(i). Let X be a set and let b_{1}, b_{2} be real-yielding many sorted sets indexed by

X. Observe that max(b_{1}, b_{2}) is real-yielding.

Let X be a set and let b_{1}, b_{2} be natural-yielding many sorted sets indexed by X. One can check that max(b_{1}, b_{2}) is natural-yielding.

One can prove the following proposition

(18) For every set X and for all real-yielding finite-support many sorted sets

b_{1}, b_{2} indexed by X holds supportmax(b_{1}, b_{2}) ⊆ supportb_{1} ∪ supportb_{2}. Let X be a set and let b_{1}, b_{2} be real-yielding finite-support many sorted sets

indexed by X. Observe that max(b_{1}, b_{2}) is finite-support.

Let A be a set and let b be a bag of A. The functor ∏ b yields a natural number and is defined by:

(Def. 5) There exists a finite sequence f of elements of ℕ such that ∏ b = ∏ f

and f = b · CFS(supportb). Let A be a set and let b be a bag of A. Then ∏ b is a natural number. One can prove the following proposition

(19) For every set X and for all bags a, b of X such that supporta misses

supportb holds ∏(a + b) = ∏ a · ∏ b. Let X be a set, let b be a real-yielding many sorted set indexed by X, and

let n be a non empty natural number. The functor b^{n} yielding a many sorted set indexed by X is defined by:

(Def. 6) support(b^{n}) = supportb and for every set i holds b^{n}(i) = b(i)^{n}.

Let X be a set, let b be a natural-yielding many sorted set indexed by X, and let n be a non empty natural number. One can verify that b^{n} is natural-yielding.

Let X be a set, let b be a real-yielding finite-support many sorted set indexed by X, and let n be a non empty natural number. Observe that b^{n} is finitesupport.

The following proposition is true

(20) For every set A holds ∏ EmptyBagA = 1.

3. Multiplicity of a Divisor

Let n, d be natural numbers. Let us assume that d ≠ 1 and n ≠ 0. The functor d -count(n) yields a natural number and is defined by:

(Def. 7) d^{d-count(n)} | n and d^{d-count(n)+1} ∤ n.

One can prove the following propositions:

(21) If n ≠ 1, then n -count(1) = 0.

(22) If 1 < n, then n -count(n) = 1.

(23) If b ≠ 0 and b < a and a ≠ 1, then a -count(b) = 0.

(24) If a ≠ 1 and a ≠ p, then a -count(p) = 0.

(25) If 1 < b, then b -count(b^{a}) = a.

(26) If b ≠ 1 and a ≠ 0 and b | b^{b-count(a)}, then b | a.

(27) If b ≠ 1, then a ≠ 0 and b -count(a) = 0 iff b ∤ a.

(28) For all non empty natural numbers a, b holds p -count(a · b) = p -count(a) + p -count(b).

(29) For all non empty natural numbers a, b holds p^{p-count(a·b)}= p^{p-count(a)} · p^{p-count(b)}.

(30) For all non empty natural numbers a, b such that b | a holds p -count(b) p -count(a).

(31) For all non empty natural numbers a, b such that b | a holds p -count(a÷ b) = p -count(a) − ′p -count(b).

(32) For every non empty natural number a holds p -count(a^{b}) = b · p -count(a).

4. Exponents in Prime-Power Factorization

Let n be a natural number. The functor PrimeExponents(n) yields a many sorted set indexed by Prime and is defined as follows:

(Def. 8) For every prime number p holds (PrimeExponents(n))(p) = p -count(n).

We introduce PFExp(n) as a synonym of PrimeExponents(n).

One can prove the following three propositions:

(33) For every set x such that x ∈ domPFExp(n) holds x is a prime number.

(34) For every set x such that x ∈ supportPFExp(n) holds x is a prime number.

(35) If a > n and n ≠ 0, then (PFExp(n))(a) = 0.

Let n be a natural number. Note that PFExp(n) is natural-yielding.

One can prove the following two propositions:

(36) If a ∈ supportPFExp(b), then a | b.

(37) If b is non empty and a is a prime number and a | b, then a ∈

supportPFExp(b). Let n be a non empty natural number. Observe that PFExp(n) is finite-

support.

We now state two propositions:

(38) For every non empty natural number a such that p | a holds (PFExp(a))(p) ≠ 0.

(39) PFExp(1) = EmptyBagPrime.

One can verify that supportPFExp(1) is empty.

One can prove the following four propositions:

(40) (PFExp(p^{a}))(p) = a.

(41) (PFExp(p))(p) = 1.

(42) If a ≠ 0, then supportPFExp(p^{a}) = {p}.

(43) supportPFExp(p) = {p}.

Let p be a prime number and let a be a non empty natural number. Observe that supportPFExp(p^{a}) is non empty and trivial.

Let p be a prime number. Observe that supportPFExp(p) is non empty and trivial.

Next we state several propositions:

(44) For all non empty natural numbers a, b such that a and b are relative prime holds supportPFExp(a) misses supportPFExp(b).

(45) For all non empty natural numbers a, b holds supportPFExp(a) ⊆ supportPFExp(a · b).

(46) For all non empty natural numbers a, b holds supportPFExp(a · b) = supportPFExp(a) ∪ supportPFExp(b).

(47) For all non empty natural numbers a, b such that a and b are relative prime holds cardsupportPFExp(a · b) = cardsupportPFExp(a) + cardsupportPFExp(b).

(48) For all non empty natural numbers a, b holds supportPFExp(a) =

supportPFExp(a^{b}). In the sequel n, m are non empty natural numbers. Next we state several propositions:

(49) PFExp(n · m) = PFExp(n) + PFExp(m).

(50) If m | n, then PFExp(n ÷ m) = PFExp(n) − ′PFExp(m).

(51) PFExp(n^{a}) = a · PFExp(n).

(52) If supportPFExp(n) = ∅, then n = 1.

(53) For all non empty natural numbers m, n holds PFExp(gcd(n, m)) = min(PFExp(n), PFExp(m)).

(54) For all non empty natural numbers m, n holds PFExp(lcm(n, m)) = max(PFExp(n), PFExp(m)).

5. Prime-Power Factorization

Let n be a non empty natural number. The functor PrimeFactorization(n) yielding a many sorted set indexed by Prime is defined as follows:

(Def. 9) supportPrimeFactorization(n) = supportPFExp(n) and for every natural number p such that p ∈ supportPFExp(n) holds (PrimeFactorization(n))(p) = p^{p-count(n)}.

We introduce PPF(n) as a synonym of PrimeFactorization(n).

Let n be a non empty natural number. Observe that PPF(n) is naturalyielding and finite-support.

The following propositions are true:

(55) If p -count(n) = 0, then (PPF(n))(p) = 0.

(56) If p -count(n) ≠ 0, then (PPF(n))(p) = p^{p-count(n)}.

(57) If supportPPF(n) = ∅, then n = 1.

(58) For all non empty natural numbers a, b such that a and b are relative prime holds PPF(a · b) = PPF(a) + PPF(b).

(59) (PPF(p^{n}))(p) = p^{n}.

fundamental theorem of arithmetic 185

(60) PPF(n^{m}) = (PPF(n))^{m}.

(61) ∏ PPF(n) = n.

References

[1] Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377–382, 1990.

[2] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41–46, 1990.

[3] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91–96, 1990.

[4] Grzegorz Bancerek. Sequences of ordinal numbers. Formalized Mathematics, 1(2):281– 290, 1990.

[5] Grzegorz Bancerek. Joining of decorated trees. Formalized Mathematics, 4(1):77–82, 1993.

[6] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107–114, 1990.

[7] Józef Białas. Group and field definitions. Formalized Mathematics, 1(3):433–439, 1990.

[8] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55– 65, 1990.

[9] Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153–164, 1990.

[10] Czesław Byliński. The sum and product of finite sequences of real numbers. Formalized Mathematics, 1(4):661–668, 1990.

[11] Marek Chmur. The lattice of natural numbers and the sublattice of it. The set of prime numbers. Formalized Mathematics, 2(4):453–459, 1991.

[12] Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165–167, 1990.

[13] Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35–40, 1990.

[14] Andrzej Kondracki. The Chinese Remainder Theorem. Formalized Mathematics, 6(4):573–577, 1997.

[15] Jarosław Kotowicz. Monotone real sequences. Subsequences. Formalized Mathematics, 1(3):471–475, 1990.

[16] Jarosław Kotowicz. Real sequences and basic operations on them. Formalized Mathematics, 1(2):269–272, 1990.

[17] Eugeniusz Kusak, Wojciech Leończuk, and Michał Muzalewski. Abelian groups, fields and vector spaces. Formalized Mathematics, 1(2):335–342, 1990.

[18] Rafał Kwiatek. Factorial and Newton coefficients. Formalized Mathematics, 1(5):887–890, 1990.

[19] Rafał Kwiatek and Grzegorz Zwara. The divisibility of integers and integer relative primes. Formalized Mathematics, 1(5):829–832, 1990.

[20] Takaya Nishiyama and Yasuho Mizuhara. Binary arithmetics. Formalized Mathematics, 4(1):83–86, 1993.

[21] Piotr Rudnicki. Little Bezout theorem (factor theorem). Formalized Mathematics, 12(1):49–58, 2003.

[22] Piotr Rudnicki and Andrzej Trybulec. Multivariate polynomials with arbitrary number of variables. Formalized Mathematics, 9(1):95–110, 2001.

[23] Christoph Schwarzweller and Andrzej Trybulec. The evaluation of multivariate polynomials. Formalized Mathematics, 9(2):331–338, 2001.

[24] Andrzej Trybulec. Subsets of complex numbers. To appear in Formalized Mathematics.

[25] Andrzej Trybulec. Tarski Grothendieck set theory. Formalized Mathematics, 1(1):9–11, 1990.

[26] Andrzej Trybulec. Many-sorted sets. Formalized Mathematics, 4(1):15–22, 1993.

[27] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67–71, 1990.

[28] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73–83, 1990.

Hilbert Space of Complex Sequences

Noboru Endou Gifu National College of Technology

Summary. An extension of [9]. As the example of complex norm spaces, we introduce the arithmetic addition and multiplication in the set of absolute summable complex sequences and also introduce the norm.

MML Identifier: CSSPACE2.

The papers [18], [21], [5], [17], [10], [22], [3], [4], [20], [19], [13], [11], [12], [15], [2], [1], [14], [16], [6], [8], and [7] provide the notation and terminology for this paper.

1. Hilbert Space of Complex Sequences

One can prove the following propositions:

(1) The carrier of Complexl2-Space = the set of l2-complex sequences and for every set x holds x is an element of Complexl2-Space iff x is a complex sequence and | id_{seq}(x)| | id_{seq}(x)| is summable and for every set x holds x is an element of Complexl2-Space iff x is a complex sequence and id_{seq}(x) \overline{id_{seq}(x)} is absolutely summable and 0_{Complexl2-Space} = CZeroseq and for every vector u of Complexl2-Space holds u = id_{seq}(u) and for all vectors u, v of Complexl2-Space holds u + v = id_{seq}(u) + id_{seq}(v) and for every Complex r and for every vector u of Complexl2-Space holds r · u = r id_{seq}(u) and for every vector u of Complexl2-Space holds − u = − id_{seq}(u) and id_{seq}(− u) = − id_{seq}(u) and for all vectors u, v of Complexl2-Space holds u − v = id_{seq}(u) − id_{seq}(v) and for all vectors v, w of Complexl2-Space holds | id_{seq}(v)| | id_{seq}(w)| is summable and for all vectors v, w of Complexl2-Space holds (v|w) = ∑(id_{seq}(v) \overline{id_{seq}(w)}).

(2) Let x, y, z be points of Complexl2-Space and a be a Complex. Then (x|x) = 0 iff x = 0_{Complexl2-Space} and ℜ((x|x)) 0 and ℑ((x|x)) = 0 and (x|y) = \overline{(y|x)} and ((x + y)|z) = (x|z) + (y|z) and ((a · x)|y) = a · (x|y).

One can verify that Complexl2-Space is complex unitary space-like. Next we state the proposition

(3) For every sequence v_{1} of Complexl2-Space such that v_{1} is Cauchy holds v_{1} is convergent.

Let us mention that Complexl2-Space is Hilbert.

2. Some Corollaries of Complex Sequences

Next we state a number of propositions:

(4) For all Complexes z_{1}, z_{2} such that ℜ(z_{1}) · ℑ(z_{2}) = ℜ(z_{2}) · ℑ(z_{1}) and ℜ(z_{1}) · ℜ(z_{2}) + ℑ(z_{1}) · ℑ(z_{2}) 0 holds |z_{1} + z_{2}| = |z_{1}| + |z_{2}|.

(5) For all Complexes x, y holds 2 · |x · y| |x|^{2} + |y|^{2}.

(6) For all Complexes x, y holds |x + y| · |x + y| 2 · |x| · |x| + 2 · |y| · |y| and |x| · |x| 2 · |x − y| · |x − y| + 2 · |y| · |y|.

(7) For every complex sequence s_{1} holds s_{1} = \overline{\overline{s_{1}}} .

(8) For every complex sequence s_{1} holds (∑_{α=0}^{κ} \overline{s_{1}} (α))_{κ∈ℕ} = \overline{(∑_{α=0}^{κ}(s_{1})(α))_{κ∈ℕ}} .

(9) Let s_{1} be a complex sequence and n be a natural number. Suppose that for every natural number i holds ℜ(s_{1})(i) 0 and ℑ(s_{1})(i) = 0. Then |(∑_{α=0}^{κ}(s_{1})(α))_{κ∈ℕ}|(n) = (∑_{α=0}^{κ} |s_{1}|(α))_{κ∈ℕ}(n).

(10) For every complex sequence s_{1} such that s_{1} is summable holds ∑ \overline{s_{1}} = \overline{∑ s_{1}} .

(11) For every complex sequence s_{1} such that s_{1} is absolutely summable holds | ∑ s_{1}| ∑ |s_{1}|.

(12) Let s_{1} be a complex sequence. Suppose s_{1} is summable and for every natural number n holds ℜ(s_{1})(n) 0 and ℑ(s_{1})(n) = 0. Then | ∑ s_{1}| = ∑ |s_{1}|.

(13) For every complex sequence s_{1} and for every natural number n holds ℜ(s_{1} \overline{s_{1}} )(n) 0 and ℑ(s_{1} \overline{s_{1}} )(n) = 0.

(14) Let s_{1} be a complex sequence. Suppose s_{1} is absolutely summable and ∑ |s_{1}| = 0. Let n be a natural number. Then s_{1}(n) = 0_{ℂ}.

(15) For every complex sequence s_{1} holds |s_{1}| = |\overline{s_{1}} |.

(16) Let c be a Complex and s_{1} be a complex sequence. Suppose s_{1} is convergent. Let r_{1} be a sequence of real numbers. Suppose that for every natural number m holds r_{1}(m) = |s_{1}(m) − c| · |s_{1}(m) − c|. Then r_{1} is convergent and limr_{1} = | lims_{1} − c| · | lims_{1} − c|.

(17) Let c be a Complex, s_{2} be a sequence of real numbers, and s_{1} be a complex sequence. Suppose s_{1} is convergent and s_{2} is convergent. Let r_{1} be a sequence of real numbers. Suppose that for every natural number i

hilbert space of complex sequences 189

holds r_{1}(i) = |s_{1}(i) − c| · |s_{1}(i) − c| + s_{2}(i). Then r_{1} is convergent and limr_{1} = | lims_{1} − c| · | lims_{1} − c| + lims_{2}.

(18) Let c be a Complex and s_{1} be a complex sequence. Suppose s_{1} is convergent. Let r_{1} be a sequence of real numbers. Suppose that for every natural number m holds r_{1}(m) = |s_{1}(m) − c| · |s_{1}(m) − c|. Then r_{1} is convergent and limr_{1} = | lims_{1} − c| · | lims_{1} − c|.

(19) Let c be a Complex, s_{2} be a sequence of real numbers, and s_{1} be a complex sequence. Suppose s_{1} is convergent and s_{2} is convergent. Let r_{1} be a sequence of real numbers. Suppose that for every natural number i holds r_{1}(i) = |s_{1}(i) − c| · |s_{1}(i) − c| + s_{2}(i). Then r_{1} is convergent and limr_{1} = | lims_{1} − c| · | lims_{1} − c| + lims_{2}.

References

[1] Agnieszka Banachowicz and Anna Winnicka. Complex sequences. Formalized Mathematics, 4(1):121–124, 1993.

[2] Czesław Byliński. The complex numbers. Formalized Mathematics, 1(3):507–513, 1990.

[3] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55– 65, 1990.

[4] Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153–164, 1990.

[5] Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47–53, 1990.

[6] Noboru Endou. Complex linear space and complex normed space. Formalized Mathematics, 12(2):93–102, 2004.

[7] Noboru Endou. Complex linear space of complex sequences. Formalized Mathematics, 12(2):109–117, 2004.

[8] Noboru Endou. Convergent sequences in complex unitary space. Formalized Mathematics, 12(2):159–165, 2004.

[9] Noboru Endou, Yasumasa Suzuki, and Yasunari Shidama. Hilbert space of real sequences. Formalized Mathematics, 11(3):255–257, 2003.

[10] Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35–40, 1990.

[11] Jarosław Kotowicz. Convergent sequences and the limit of sequences. Formalized Mathematics, 1(2):273–275, 1990.

[12] Jarosław Kotowicz. Monotone real sequences. Subsequences. Formalized Mathematics, 1(3):471–475, 1990.

[13] Jarosław Kotowicz. Real sequences and basic operations on them. Formalized Mathematics, 1(2):269–272, 1990.

[14] Adam Naumowicz. Conjugate sequences, bounded complex sequences and convergent complex sequences. Formalized Mathematics, 6(2):265–268, 1997.

[15] Konrad Raczkowski and Andrzej Nędzusiak. Series. Formalized Mathematics, 2(4):449– 452, 1991.

[16] Yasunari Shidama and Artur Korniłowicz. Convergence and the limit of complex sequences. Series. Formalized Mathematics, 6(3):403–410, 1997.

[17] Andrzej Trybulec. Subsets of complex numbers. To appear in Formalized Mathematics.

[18] Andrzej Trybulec. Tarski Grothendieck set theory. Formalized Mathematics, 1(1):9–11, 1990.

[19] Andrzej Trybulec and Czesław Byliński. Some properties of real numbers. Formalized Mathematics, 1(3):445–449, 1990.

190 noboru endou

[20] Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291–296, [21] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67–71, 1990.

[22] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73–83, 1990.

Banach Space of Absolute Summable Complex Sequences

Noboru Endou Gifu National College of Technology

Summary. An extension of [16]. As the example of complex norm spaces, I introduced the arithmetic addition and multiplication in the set of absolute summable complex sequences and also introduced the norm.

MML Identifier: CSSPACE3.

The terminology and notation used in this paper are introduced in the following articles: [18], [20], [6], [2], [17], [9], [21], [4], [5], [19], [13], [11], [10], [14], [3], [1], [12], [15], [7], and [8].

1. Complex-L1-Space: The Space of Absolute Summable Complex Sequences

The subset the set of l1-complex sequences of the linear space of complex sequences is defined by the condition (Def. 1).

(Def. 1) Let x be a set. Then x ∈ the set of l1-complex sequences if and only if

x ∈ the set of complex sequences and id_{seq}(x) is absolutely summable. The following proposition is true

(1) Let c be a Complex, s_{1} be a complex sequence, and r_{1} be a sequence of real numbers. Suppose s_{1} is convergent and for every natural number i holds r_{1}(i) = |s_{1}(i) − c|. Then r_{1} is convergent and limr_{1} = | lims_{1} − c|.

Let us note that the set of l1-complex sequences is non empty. Let us observe that the set of l1-complex sequences is linearly closed. Next we state the proposition (2) ⟨the set of l1-complex sequences, Zero (the set of l1-complex

sequences, the linear space of complex sequences), Add (the set of l1complex sequences, the linear space of complex sequences), Mult (the set of l1-complex sequences, the linear space of complex sequences)⟩ is a subspace of the linear space of complex sequences.

Let us note that ⟨the set of l1-complex sequences, Zero (the set of l1-complex sequences, the linear space of complex sequences), Add (the set of l1-complex sequences, the linear space of complex sequences), Mult (the set of l1-complex sequences, the linear space of complex sequences)⟩ is Abelian, add-associative, right zeroed, right complementable, and complex linear space-like.

We now state the proposition

(3) ⟨the set of l1-complex sequences, Zero (the set of l1-complex sequences, the linear space of complex sequences), Add (the set of l1complex sequences, the linear space of complex sequences), Mult (the set of l1-complex sequences, the linear space of complex sequences)⟩ is a complex linear space.

The function cl norm from the set of l1-complex sequences into ℝ is defined as follows:

(Def. 2) For every set x such that x ∈ the set of l1-complex sequences holds

cl norm(x) = ∑ | id_{seq}(x)|. Let X be a non empty set, let Z be an element of X, let A be a binary

operation on X, let M be a function from [: ℂ, X:] into X, and let N be a function from X into ℝ. Note that ⟨X, Z, A, M, N⟩ is non empty.

We now state four propositions:

(4) Let l be a complex normed space structure. Suppose ⟨the carrier of l, the zero of l, the addition of l, the external multiplication of l⟩ is a complex linear space. Then l is a complex linear space.

(5) Let c_{1} be a complex sequence. Suppose that for every natural number n holds c_{1}(n) = 0_{ℂ}. Then c_{1} is absolutely summable and ∑ |c_{1}| = 0.

(6) Let c_{1} be a complex sequence. Suppose c_{1} is absolutely summable and ∑ |c_{1}| = 0. Let n be a natural number. Then c_{1}(n) = 0_{ℂ}.

(7) ⟨the set of l1-complex sequences, Zero (the set of l1-complex sequences, the linear space of complex sequences), Add (the set of l1complex sequences, the linear space of complex sequences), Mult (the set of l1-complex sequences, the linear space of complex sequences), cl norm⟩ is a complex linear space.

The non empty complex normed space structure Complex-l1-Space is defined by the condition (Def. 3).

(Def. 3) Complex-l1-Space = ⟨the set of l1-complex sequences, Zero (the set of l1complex sequences, the linear space of complex sequences), Add (the set of l1-complex sequences, the linear space of complex sequences), Mult (the set of l1-complex sequences, the linear space of complex sequences), cl norm⟩.

2. Complex-L1-Space is Banach

One can prove the following propositions:

(8) The carrier of Complex-l1-Space = the set of l1-complex sequences and for every set x holds x is a vector of Complex-l1-Space iff x is a complex sequence and id_{seq}(x) is absolutely summable and 0_{Complex-l1-Space} = CZeroseq and for every vector u of Complex-l1-Space holds u = id_{seq}(u) and for all vectors u, v of Complex-l1-Space holds u+v = id_{seq}(u)+id_{seq}(v) and for every Complex p and for every vector u of Complex-l1-Space holds p · u = p id_{seq}(u) and for every vector u of Complex-l1-Space holds − u = − id_{seq}(u) and id_{seq}(− u) = − id_{seq}(u) and for all vectors u, v of Complex-l1-Space holds u − v = id_{seq}(u) − id_{seq}(v) and for every vector v of Complex-l1-Space holds id_{seq}(v) is absolutely summable and for every vector v of Complex-l1-Space holds ∥v∥ = ∑ | id_{seq}(v)|.

(9) Let x, y be points of Complex-l1-Space and p be a Complex. Then ∥x∥ = 0 iff x = 0_{Complex-l1-Space} and 0 ∥x∥ and ∥x + y∥ ∥x∥ + ∥y∥ and ∥p · x∥ = |p| · ∥x∥.

Let us observe that Complex-l1-Space is complex normed space-like, complex linear space-like, Abelian, add-associative, right zeroed, and right complementable.

Let X be a non empty complex normed space structure and let x, y be points of X. The functor ρ(x, y) yielding a real number is defined as follows:

(Def. 4) ρ(x, y) = ∥x − y∥.

Let C_{1} be a non empty complex normed space structure and let s_{2} be a sequence of C_{1}. We say that s_{2} is CCauchy if and only if the condition (Def. 5) is satisfied.

(Def. 5) Let r_{2} be a real number. Suppose r_{2} > 0. Then there exists a natural number k_{1} such that for all natural numbers n_{1}, m_{1} if n_{1} k_{1} and m_{1} k_{1}, then ρ(s_{2}(n_{1}), s_{2}(m_{1})) < r_{2}.

We introduce s_{1} is Cauchy sequence by norm as a synonym of s_{2} is CCauchy.

In the sequel N_{1} is a non empty complex normed space and s_{1} is a sequence of N_{1}.

One can prove the following propositions:

(10) s_{1} is Cauchy sequence by norm if and only if for every real number r such that r > 0 there exists a natural number k such that for all natural numbers n, m such that n k and m k holds ∥s_{1}(n) − s_{1}(m)∥ < r.

(11) For every sequence v_{1} of Complex-l1-Space such that v_{1} is Cauchy sequence by norm holds v_{1} is convergent.

References

[1] Agnieszka Banachowicz and Anna Winnicka. Complex sequences. Formalized Mathematics, 4(1):121–124, 1993.

[2] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91–96, 1990.

[3] Czesław Byliński. The complex numbers. Formalized Mathematics, 1(3):507–513, 1990.

[4] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55– 65, 1990.

[5] Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153–164, 1990.

[6] Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47–53, 1990.

[7] Noboru Endou. Complex linear space and complex normed space. Formalized Mathematics, 12(2):93–102, 2004.

[8] Noboru Endou. Complex linear space of complex sequences. Formalized Mathematics, 12(2):109–117, 2004.

[9] Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35–40, 1990.

[10] Jarosław Kotowicz. Convergent sequences and the limit of sequences. Formalized Mathematics, 1(2):273–275, 1990.

[11] Jarosław Kotowicz. Real sequences and basic operations on them. Formalized Mathematics, 1(2):269–272, 1990.

[12] Adam Naumowicz. Conjugate sequences, bounded complex sequences and convergent complex sequences. Formalized Mathematics, 6(2):265–268, 1997.

[13] Jan Popiołek. Real normed space. Formalized Mathematics, 2(1):111–115, 1991.

[14] Konrad Raczkowski and Andrzej Nędzusiak. Series. Formalized Mathematics, 2(4):449– 452, 1991.

[15] Yasunari Shidama and Artur Korniłowicz. Convergence and the limit of complex sequences. Series. Formalized Mathematics, 6(3):403–410, 1997.

[16] Yasumasa Suzuki, Noboru Endou, and Yasunari Shidama. Banach space of absolute summable real sequences. Formalized Mathematics, 11(4):377–380, 2003.

[17] Andrzej Trybulec. Subsets of complex numbers. To appear in Formalized Mathematics.

[18] Andrzej Trybulec. Tarski Grothendieck set theory. Formalized Mathematics, 1(1):9–11, 1990.

[19] Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291–296, 1990.

[20] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67–71, 1990.

[21] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73–83, 1990.

The Taylor Expansions

Yasunari Shidama Shinshu University Nagano

Summary. In this article, some classic theorems of calculus are described. The Taylor expansions and the logarithmic differentiation, etc. are included here.

MML Identifier: TAYLOR 1.

The terminology and notation used in this paper have been introduced in the following articles: [22], [24], [25], [4], [6], [9], [5], [11], [20], [18], [3], [8], [2], [21], [7], [1], [23], [14], [12], [10], [17], [19], [13], [15], [16], and [26].

1. The Logarithmic Differentiation Method

For simplicity, we use the following convention: n denotes a natural number, i denotes an integer, p, x, x_{0}, y denote real numbers, q denotes a rational number, and f denotes a partial function from ℝ to ℝ.

Let q be an integer. The functor _{ℤ}^{q} yields a function from ℝ into ℝ and is defined as follows:

(Def. 1) For every real number x holds (_{ℤ}^{q})(x) = x_{ℤ}^{q}. Next we state a number of propositions:

(1) For all natural numbers m, n holds x_{ℤ}^{n+m} = (x_{ℤ}^{n}) · x_{ℤ}^{m}.

(2) _{ℤ}^{n} is differentiable in x and (_{ℤ}^{n})′(x) = n · x_{ℤ}^{n−1} .

(3) If f is differentiable in x_{0}, then (_{ℤ}^{n}) · f is differentiable in x_{0} and ((_{ℤ}^{n}) · f)′(x_{0}) = n · f(x_{0})_{ℤ}^{n−1} · f′(x_{0}).

(4) exp(− x) = \frac{1}{exp x}.

(5) (expx)ℝ^{\frac{1}{i}} = exp(\frac{x}{i}).

(6) For all integers m, n holds (expx)ℝ^{\frac{m}{n}} = exp(\frac{m}{n} · x).

(7) (expx)_{ℚ}^{q} = exp(q · x).

(8) (expx)_{ℝ}^{p} = exp(p · x).

(9) (exp1)_{ℝ}^{x} = expx and (exp1)^{x} = expx and e^{x} = expx and e_{ℝ}^{x} = expx.

(10) exp(1)_{ℝ}^{x} = exp(x) and exp(1)^{x} = exp(x) and e^{x} = exp(x) and e_{ℝ}^{x} = exp(x).

(11) e 2.

(12) log_{e} expx = x.

(13) log_{e} exp(x) = x.

(14) If y > 0, then explog_{e} y = y.

(15) If y > 0, then exp(log_{e} y) = y.

(16) exp is one-to-one and exp is differentiable on ℝ and exp is differentiable on Ω_{ℝ} and for every real number x holds exp′(x) = exp(x) and for every real number x holds 0 < exp′(x) and domexp = ℝ and domexp = Ω_{ℝ} and rngexp = ]0, +∞[.

Let us note that exp is one-to-one.

We now state the proposition

(17) exp^{−1} is differentiable on dom(exp^{−1} ) and for every real number x such

that x ∈ dom(exp^{−1} ) holds (exp^{−1} )′(x) = \frac{1}{x}. Let us mention that ]0, +∞[ is non empty. Let a be a real number. The functor log (a) yields a partial function from ℝ

to ℝ and is defined by: (Def. 2) domlog (a) = ]0, +∞[ and for every element d of ]0, +∞[ holds

(log (a))(d) = log_{a} d. One can prove the following three propositions:

(18) log (e) = exp^{−1} and log (e) is one-to-one and domlog (e) = ]0, +∞[ and rnglog (e) = ℝ and log (e) is differentiable on ]0, +∞[ and for every real number x such that x > 0 holds log (e) is differentiable in x and for every element x of ]0, +∞[ holds (log (e))′(x) = \frac{1}{x} and for every element x of ]0, +∞[ holds 0 < (log (e))′(x).

(19) If f is differentiable in x_{0}, then exp·f is differentiable in x_{0} and (exp·f)′(x_{0}) = exp(f(x_{0})) · f′(x_{0}).

(20) If f is differentiable in x_{0} and f(x_{0}) > 0, then log (e) · f is differentiable

in x_{0} and (log (e) · f)′(x_{0}) = \frac{f′(x_{0})}{f(x_{0})}. Let p be a real number. The functor _{ℝ}^{p} yielding a partial function from ℝ to

ℝ is defined as follows: (Def. 3) dom(_{ℝ}^{p}) = ]0, +∞[ and for every element d of ]0, +∞[ holds (_{ℝ}^{p})(d) = d_{ℝ}^{p}.

We now state two propositions:

(21) If x > 0, then _{ℝ}^{p} is differentiable in x and (_{ℝ}^{p})′(x) = p · x_{ℝ}^{p−1} .

(22) If f is differentiable in x_{0} and f(x_{0}) > 0, then (_{ℝ}^{p}) · f is differentiable in x_{0} and ((_{ℝ}^{p}) · f)′(x_{0}) = p · f(x_{0})_{ℝ}^{p−1} · f′(x_{0}).

2. The Taylor Expansions

Let f be a partial function from ℝ to ℝ and let Z be a subset of ℝ. The functor f′(Z) yields a sequence of partial functions from ℝ into ℝ and is defined by:

(Def. 4) f′(Z)(0) = f↾Z and for every natural number i holds f′(Z)(i + 1) =

f′(Z)(i)′_{↾Z}. Let f be a partial function from ℝ to ℝ, let n be a natural number, and let

Z be a subset of ℝ. We say that f is differentiable n times on Z if and only if: (Def. 5) For every natural number i such that i n − 1 holds f′(Z)(i) is diffe-

rentiable on Z. The following proposition is true

(23) Let f be a partial function from ℝ to ℝ, Z be a subset of ℝ, and n be a natural number. Suppose f is differentiable n times on Z. Let m be a natural number. If m n, then f is differentiable m times on Z.

Let f be a partial function from ℝ to ℝ, let Z be a subset of ℝ, and let a, b be real numbers. The functor Taylor(f, Z, a, b) yields a sequence of real numbers and is defined as follows:

(Def. 6) For every natural number n holds (Taylor(f, Z, a, b))(n) = \frac{f′(Z)(n)(a)·(b−a)^{n}}{n!} .

The following propositions are true:

(24) Let f be a partial function from ℝ to ℝ, Z be a subset of ℝ, and n be a natural number. Suppose f is differentiable n times on Z. Let a, b be real numbers. If a < b and ]a, b[ ⊆ Z, then f′(Z)(n)↾]a, b[ = f′(]a, b[)(n).

(25) Let n be a natural number, f be a partial function from ℝ to ℝ, and Z be a subset of ℝ. Suppose f is differentiable n times on Z. Let a, b be real numbers. Suppose a < b and [a, b] ⊆ Z and f′(Z)(n) is continuous on [a, b] and f is differentiable n + 1 times on ]a, b[. Let l be a real number and g be a partial function from ℝ to ℝ. Suppose domg = ℝ and for every real number x holds g(x) = f(b) − (∑_{α=0}^{κ}(Taylor(f, Z, x, b))(α))_{κ∈ℕ}(n) − \frac{l·(b−x)^{n+1}}{(n+1)!} and f(b) − (∑_{α=0}^{κ}(Taylor(f, Z, a, b))(α))_{κ∈ℕ}(n) − \frac{l·(b−a)^{n+1}}{(n+1)!} = 0. Then

(i) g is differentiable on ]a, b[, (ii) g(a) = 0, (iii) g(b) = 0, (iv) g is continuous on [a, b], and

(v) for every real number x such that x ∈ ]a, b[ holds g′(x) = − \frac{f′(]a,b[)(n+1)(x)·(b−x)^{n}}{n!} + \frac{l·(b−x)^{n}}{n!} .

(26) Let n be a natural number, f be a partial function from ℝ to ℝ, Z be a subset of ℝ, and b, l be real numbers. Then there exists a function g from ℝ into ℝ such that for every real number x holds g(x) = f(b) − (∑_{α=0}^{κ}(Taylor(f, Z, x, b))(α))_{κ∈ℕ}(n) − \frac{l·(b−x)^{n+1}}{(n+1)!} .

(27) Let n be a natural number, f be a partial function from ℝ to ℝ, and Z be a subset of ℝ. Suppose f is differentiable n times on Z. Let a, b be real numbers. Suppose a < b and [a, b] ⊆ Z and f′(Z)(n) is continuous on [a, b] and f is differentiable n+1 times on ]a, b[. Then there exists a real number c such that c ∈ ]a, b[ and f(b) = (∑_{α=0}^{κ}(Taylor(f, Z, a, b))(α))_{κ∈ℕ}(n) + \frac{f′(]a,b[)(n+1)(c)·(b−a)^{n+1}}{(n+1)!} .

(28) Let n be a natural number, f be a partial function from ℝ to ℝ, and Z be a subset of ℝ. Suppose f is differentiable n times on Z. Let a, b be real numbers. Suppose a < b and [a, b] ⊆ Z and f′(Z)(n) is continuous on [a, b] and f is differentiable n + 1 times on ]a, b[. Let l be a real number and g be a partial function from ℝ to ℝ. Suppose domg = ℝ and for every real number x holds g(x) = f(a) − (∑_{α=0}^{κ}(Taylor(f, Z, x, a))(α))_{κ∈ℕ}(n) − \frac{l·(a−x)^{n+1}}{(n+1)!} and f(a) − (∑_{α=0}^{κ}(Taylor(f, Z, b, a))(α))_{κ∈ℕ}(n) − \frac{l·(a−b)^{n+1}}{(n+1)!} = 0. Then

(i) g is differentiable on ]a, b[, (ii) g(b) = 0, (iii) g(a) = 0, (iv) g is continuous on [a, b], and

(v) for every real number x such that x ∈ ]a, b[ holds g′(x) = − \frac{f′(]a,b[)(n+1)(x)·(a−x)^{n}}{n!} + \frac{l·(a−x)^{n}}{n!} .

(29) Let n be a natural number, f be a partial function from ℝ to ℝ, and Z be a subset of ℝ. Suppose f is differentiable n times on Z. Let a, b be real numbers. Suppose a < b and [a, b] ⊆ Z and f′(Z)(n) is continuous on [a, b] and f is differentiable n+1 times on ]a, b[. Then there exists a real number c such that c ∈ ]a, b[ and f(a) = (∑_{α=0}^{κ}(Taylor(f, Z, b, a))(α))_{κ∈ℕ}(n) + \frac{f′(]a,b[)(n+1)(c)·(a−b)^{n+1}}{(n+1)!} .

(30) Let f be a partial function from ℝ to ℝ, Z be a subset of ℝ, and Z_{1} be an open subset of ℝ. Suppose Z_{1} ⊆ Z. Let n be a natural number. If f is differentiable n times on Z, then f′(Z)(n)↾Z_{1} = f′(Z_{1})(n).

(31) Let f be a partial function from ℝ to ℝ, Z be a subset of ℝ, and Z_{1} be an open subset of ℝ. Suppose Z_{1} ⊆ Z. Let n be a natural number. Suppose f is differentiable n + 1 times on Z. Then f is differentiable n + 1 times on Z_{1}.

(32) Let f be a partial function from ℝ to ℝ, Z be a subset of ℝ, and x be a real number. If x ∈ Z, then for every natural number n holds f(x) = (∑_{α=0}^{κ}(Taylor(f, Z, x, x))(α))_{κ∈ℕ}(n).

the taylor expansions 199

(33) Let n be a natural number, f be a partial function from ℝ to ℝ, and x_{0}, r be real numbers. Suppose 0 < r and f is differentiable n + 1 times on ]x_{0} − r, x_{0} + r[. Let x be a real number. Suppose x ∈ ]x_{0} − r, x_{0} + r[. Then there exists a real number s such that 0 < s and s < 1 and f(x) = (∑_{α=0}^{κ}(Taylor(f,]x_{0} − r, x_{0} + r[, x_{0}, x))(α))_{κ∈ℕ}(n) + \frac{f′(]x_{0}−r,x_{0}+r[)(n+1)(x_{0}+s·(x−x_{0}))·(x−x_{0})^{n+1}}{(n+1)!} .

References

[1] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41–46, 1990.

[2] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91–96, 1990.

[3] Grzegorz Bancerek. Sequences of ordinal numbers. Formalized Mathematics, 1(2):281– 290, 1990.

[4] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55– 65, 1990.

[5] Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153–164, 1990.

[6] Czesław Byliński. Partial functions. Formalized Mathematics, 1(2):357–367, 1990.

[7] Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35–40, 1990.

[8] Andrzej Kondracki. Basic properties of rational numbers. Formalized Mathematics, 1(5):841–845, 1990.

[9] Jarosław Kotowicz. Partial functions from a domain to a domain. Formalized Mathematics, 1(4):697–702, 1990.

[10] Jarosław Kotowicz. Real sequences and basic operations on them. Formalized Mathematics, 1(2):269–272, 1990.

[11] Jarosław Kotowicz. The limit of a real function at infinity. Formalized Mathematics, 2(1):17–28, 1991.

[12] Rafał Kwiatek. Factorial and Newton coefficients. Formalized Mathematics, 1(5):887–890, 1990.

[13] Beata Perkowska. Functional sequence from a domain to a domain. Formalized Mathematics, 3(1):17–21, 1992.

[14] Jan Popiołek. Some properties of functions modul and signum. Formalized Mathematics, 1(2):263–264, 1990.

[15] Konrad Raczkowski. Integer and rational exponents. Formalized Mathematics, 2(1):125– 130, 1991.

[16] Konrad Raczkowski and Andrzej Nędzusiak. Real exponents and logarithms. Formalized Mathematics, 2(2):213–216, 1991.

[17] Konrad Raczkowski and Andrzej Nędzusiak. Series. Formalized Mathematics, 2(4):449– 452, 1991.

[18] Konrad Raczkowski and Paweł Sadowski. Real function continuity. Formalized Mathematics, 1(4):787–791, 1990.

[19] Konrad Raczkowski and Paweł Sadowski. Real function differentiability. Formalized Mathematics, 1(4):797–801, 1990.

[20] Konrad Raczkowski and Paweł Sadowski. Topological properties of subsets in real numbers. Formalized Mathematics, 1(4):777–780, 1990.

[21] Andrzej Trybulec. Subsets of complex numbers. To appear in Formalized Mathematics.

[22] Andrzej Trybulec. Tarski Grothendieck set theory. Formalized Mathematics, 1(1):9–11, 1990.

[23] Michał J. Trybulec. Integers. Formalized Mathematics, 1(3):501–505, 1990.

[24] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67–71, 1990.

[25] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73–83, 1990.

200 yasunari shidama

[26] Yuguang Yang and Yasunari Shidama. Trigonometric functions and existence of circle

ratio. Formalized Mathematics, 7(2):255–263, 1998.

Received February 24, 2004

Complex Banach Space of Bounded Linear Operators

Noboru Endou Gifu National College of Technology

Summary. An extension of [19]. In this article, the basic properties of complex linear spaces which are defined by the set of all complex linear operators from one complex linear space to another are described. Finally, a complex Banach space is introduced. This is defined by the set of all bounded complex linear operators, like in [19].

MML Identifier: CLOPBAN1.

The articles [24], [6], [26], [27], [4], [5], [17], [22], [21], [2], [1], [20], [11], [7], [25], [23], [18], [15], [13], [14], [12], [16], [3], [9], [10], [8], and [19] provide the terminology and notation for this paper.

1. Complex Vector Space of Operators

Let X be a set, let Y be a non empty set, let F be a function from [: ℂ, Y :] into Y, let c be a complex number, and let f be a function from X into Y. Then F^{∘}(c, f) is an element of Y^{X}.

We now state the proposition

(1) Let X be a non empty set and Y be a complex linear space. Then there exists a function M_{1} from [: ℂ, (the carrier of Y)^{X} :] into (the carrier of Y)^{X} such that for every Complex c and for every element f of (the carrier of Y)^{X} and for every element s of X holds M_{1}(⟨⟨c, f⟩⟩)(s) = c · f(s).

Let X be a non empty set and let Y be a complex linear space. The functor FuncExtMult(X, Y) yields a function from [: ℂ, (the carrier of Y)^{X} :] into (the carrier of Y)^{X} and is defined by the condition (Def. 1).

(Def. 1) Let c be a Complex, f be an element of (the carrier of Y)^{X}, and x be an element of X. Then (FuncExtMult(X, Y))(⟨⟨c, f⟩⟩)(x) = c · f(x).

We follow the rules: X is a non empty set, Y is a complex linear space, and f, g, h are elements of (the carrier of Y)^{X}.

We now state the proposition

(2) For every element x of X holds (FuncZero(X, Y))(x) = 0_{Y} . In the sequel a, b are Complexes. Next we state several propositions:

(3) h = (FuncExtMult(X, Y))(⟨⟨a, f⟩⟩) iff for every element x of X holds h(x) = a · f(x).

(4) (FuncAdd(X, Y))(f, g) = (FuncAdd(X, Y))(g, f).

(5) (FuncAdd(X, Y))(f, (FuncAdd(X, Y))(g, h)) = (FuncAdd(X, Y))((FuncAdd(X, Y))(f, g), h).

(6) (FuncAdd(X, Y))(FuncZero(X, Y), f) = f.

(7) (FuncAdd(X, Y))(f, (FuncExtMult(X, Y))(⟨⟨− 1_{ℂ}, f⟩⟩)) = FuncZero(X, Y).

(8) (FuncExtMult(X, Y))(⟨⟨1_{ℂ}, f⟩⟩) = f.

(9) (FuncExtMult(X, Y))(⟨⟨a, (FuncExtMult(X, Y))(⟨⟨b, f⟩⟩)⟩⟩) = (FuncExtMult(X, Y))(⟨⟨a · b, f⟩⟩).

(10) (FuncAdd(X, Y))((FuncExtMult(X, Y))(⟨⟨a, f⟩⟩), (FuncExtMult(X, Y))(⟨⟨b, f⟩⟩)) = (FuncExtMult(X, Y))(⟨⟨a + b, f⟩⟩).

(11) ⟨(the carrier of Y)^{X}, FuncZero(X, Y), FuncAdd(X, Y),

FuncExtMult(X, Y)⟩ is a complex linear space. Let X be a non empty set and let Y be a complex linear space. The functor

ComplexVectSpace(X, Y) yielding a complex linear space is defined as follows: (Def. 2) ComplexVectSpace(X, Y) = ⟨(the carrier of Y)^{X}, FuncZero(X, Y),

FuncAdd(X, Y), FuncExtMult(X, Y)⟩. Let X be a non empty set and let Y be a complex linear space. Observe that

ComplexVectSpace(X, Y) is strict.

Let X be a non empty set and let Y be a complex linear space. Observe that every vector of ComplexVectSpace(X, Y) is function-like and relation-like.

Let X be a non empty set, let Y be a complex linear space, let f be a vector of ComplexVectSpace(X, Y), and let x be an element of X. Then f(x) is a vector of Y.

We now state three propositions:

(12) Let X be a non empty set, Y be a complex linear space, and f, g, h be vectors of ComplexVectSpace(X, Y). Then h = f + g if and only if for every element x of X holds h(x) = f(x) + g(x).

(13) Let X be a non empty set, Y be a complex linear space, f, h be vectors of ComplexVectSpace(X, Y), and c be a Complex. Then h = c · f if and only if for every element x of X holds h(x) = c · f(x).

(14) For every non empty set X and for every complex linear space Y holds 0_{ComplexVectSpace (X,Y)} = X ⟼→ 0_{Y} .

2. Complex Vector Space of Linear Operators

Let X be a non empty CLS structure, let Y be a non empty loop structure, and let I_{1} be a function from X into Y. We say that I_{1} is additive if and only if:

(Def. 3) For all vectors x, y of X holds I_{1}(x + y) = I_{1}(x) + I_{1}(y).

Let X, Y be non empty CLS structures and let I_{1} be a function from X into Y. We say that I_{1} is homogeneous if and only if:

(Def. 4) For every vector x of X and for every Complex r holds I_{1}(r·x) = r·I_{1}(x).

Let X be a non empty CLS structure and let Y be a complex linear space. One can verify that there exists a function from X into Y which is additive and homogeneous.

Let X, Y be complex linear spaces. A linear operator from X into Y is an additive homogeneous function from X into Y.

Let X, Y be complex linear spaces. The functor LinearOperators(X, Y) yielding a subset of ComplexVectSpace(the carrier of X, Y) is defined by:

(Def. 5) For every set x holds x ∈ LinearOperators(X, Y) iff x is a linear operator

from X into Y. Let X, Y be complex linear spaces. Note that LinearOperators(X, Y) is non

empty.

Next we state two propositions:

(15) For all complex linear spaces X, Y holds LinearOperators(X, Y) is linearly closed.

(16) Let X, Y be complex linear spaces. Then ⟨LinearOperators(X, Y), Zero (LinearOperators(X, Y), ComplexVectSpace(the carrier of X, Y)), Add (LinearOperators(X, Y), ComplexVectSpace(the carrier of X, Y)), Mult (LinearOperators(X, Y), ComplexVectSpace(the carrier of X, Y))⟩ is a subspace of ComplexVectSpace(the carrier of X, Y).

Let X, Y be complex linear spaces. One can check that

⟨LinearOperators(X, Y), Zero (LinearOperators(X, Y), ComplexVectSpace(the carrier of X, Y)), Add (LinearOperators(X, Y), ComplexVectSpace(the carrier of X, Y)), Mult (LinearOperators(X, Y), ComplexVectSpace(the carrier of X, Y))⟩ is Abelian, add-associative, right zeroed, right complementable, and complex linear space-like.

Next we state the proposition

(17) Let X, Y be complex linear spaces. Then ⟨LinearOperators(X, Y), Zero (LinearOperators(X, Y), ComplexVectSpace(the carrier of X, Y)), Add (LinearOperators(X, Y), ComplexVectSpace(the carrier of X, Y)), Mult (LinearOperators(X, Y), ComplexVectSpace(the carrier of X, Y))⟩ is a complex linear space.

Let X, Y be complex linear spaces. The functor CVSpLinOps(X, Y) yielding a complex linear space is defined as follows:

(Def. 6) CVSpLinOps(X, Y) = ⟨LinearOperators(X, Y), Zero (LinearOperators (X, Y), ComplexVectSpace(the carrier of X, Y)), Add (LinearOperators(X, Y), ComplexVectSpace(the carrier of X, Y)), Mult (LinearOperators(X, Y), ComplexVectSpace(the carrier of X, Y))⟩.

Let X, Y be complex linear spaces. Note that CVSpLinOps(X, Y) is strict. Let X, Y be complex linear spaces. One can check that every element of

CVSpLinOps(X, Y) is function-like and relation-like.

Let X, Y be complex linear spaces, let f be an element of CVSpLinOps(X, Y), and let v be a vector of X. Then f(v) is a vector of Y.

Next we state four propositions:

(18) Let X, Y be complex linear spaces and f, g, h be vectors of CVSpLinOps(X, Y). Then h = f + g if and only if for every vector x of X holds h(x) = f(x) + g(x).

(19) Let X, Y be complex linear spaces, f, h be vectors of CVSpLinOps(X, Y), and c be a Complex. Then h = c · f if and only if for every vector x of X holds h(x) = c · f(x).

(20) For all complex linear spaces X, Y holds 0_{CVSpLinOps (X,Y)} = (the carrier of X) ⟼→ 0_{Y} .

(21) For all complex linear spaces X, Y holds (the carrier of X) ⟼→ 0_{Y} is a linear operator from X into Y.

3. Complex Normed Linear Space of Bounded Linear Operators

One can prove the following proposition

(22) Let X be a complex normed space, s_{1} be a sequence of X, and g be a point of X. If s_{1} is convergent and lims_{1} = g, then ∥s_{1}∥ is convergent and lim∥s_{1}∥ = ∥g∥.

Let X, Y be complex normed spaces and let I_{1} be a linear operator from X into Y. We say that I_{1} is bounded if and only if:

(Def. 7) There exists a real number K such that 0 K and for every vector x of

X holds ∥I_{1}(x)∥ K · ∥x∥. We now state the proposition

(23) Let X, Y be complex normed spaces and f be a linear operator from X into Y. If for every vector x of X holds f(x) = 0_{Y} , then f is bounded.

Let X, Y be complex normed spaces. Observe that there exists a linear operator from X into Y which is bounded.

Let X, Y be complex normed spaces. The functor BdLinOps(X, Y) yielding a subset of CVSpLinOps(X, Y) is defined as follows:

(Def. 8) For every set x holds x ∈ BdLinOps(X, Y) iff x is a bounded linear

operator from X into Y. Let X, Y be complex normed spaces. One can check that BdLinOps(X, Y)

is non empty.

One can prove the following two propositions:

(24) For all complex normed spaces X, Y holds BdLinOps(X, Y) is linearly closed.

(25) For all complex normed spaces X, Y holds ⟨BdLinOps(X, Y), Zero (BdLinOps(X, Y), CVSpLinOps(X, Y)), Add (BdLinOps(X, Y), CVSpLinOps(X, Y)), Mult (BdLinOps(X, Y), CVSpLinOps(X, Y))⟩ is a subspace of CVSpLinOps(X, Y).

Let X, Y be complex normed spaces. Observe that ⟨BdLinOps(X, Y), Zero (BdLinOps(X, Y), CVSpLinOps(X, Y)), Add (BdLinOps(X, Y), CVSpLinOps(X, Y)), Mult (BdLinOps(X, Y), CVSpLinOps(X, Y))⟩ is Abe-

lian, add-associative, right zeroed, right complementable, and complex linear space-like.

Next we state the proposition

(26) For all complex normed spaces X, Y holds ⟨BdLinOps(X, Y), Zero (BdLinOps(X, Y), CVSpLinOps(X, Y)), Add (BdLinOps(X, Y), CVSpLinOps(X, Y)), Mult (BdLinOps(X, Y), CVSpLinOps(X, Y))⟩ is a complex linear space.

Let X, Y be complex normed spaces. The functor CVSpBdLinOps(X, Y) yielding a complex linear space is defined by:

(Def. 9) CVSpBdLinOps(X, Y) = ⟨BdLinOps(X, Y), Zero (BdLinOps(X, Y), CVSpLinOps(X, Y)), Add (BdLinOps(X, Y), CVSpLinOps(X, Y)), Mult (BdLinOps(X, Y), CVSpLinOps(X, Y))⟩.

Let X, Y be complex normed spaces. One can check that CVSpBdLinOps(X, Y) is strict.

Let X, Y be complex normed spaces. Note that every element of CVSpBdLinOps(X, Y) is function-like and relation-like.

Let X, Y be complex normed spaces, let f be an element of CVSpBdLinOps(X, Y), and let v be a vector of X. Then f(v) is a vector of Y.

One can prove the following propositions:

(27) Let X, Y be complex normed spaces and f, g, h be vectors of CVSpBdLinOps(X, Y). Then h = f + g if and only if for every vector

x of X holds h(x) = f(x) + g(x).

(28) Let X, Y be complex normed spaces, f, h be vectors of CVSpBdLinOps(X, Y), and c be a Complex. Then h = c · f if and only if for every vector x of X holds h(x) = c · f(x).

(29) For all complex normed spaces X, Y holds 0_{CVSpBdLinOps (X,Y)} = (the

carrier of X) ⟼→ 0_{Y} . Let X, Y be complex normed spaces and let f be a set. Let us assume that

f ∈ BdLinOps(X, Y). The functor modetrans(f, X, Y) yields a bounded linear operator from X into Y and is defined as follows:

(Def. 10) modetrans(f, X, Y) = f.

Let X, Y be complex normed spaces and let u be a linear operator from X into Y. The functor PreNorms(u) yielding a non empty subset of ℝ is defined as follows:

(Def. 11) PreNorms(u) = {∥u(t)∥; t ranges over vectors of X: ∥t∥ 1}.

We now state three propositions:

(30) Let X, Y be complex normed spaces and g be a bounded linear operator from X into Y. Then PreNorms(g) is non empty and upper bounded.

(31) Let X, Y be complex normed spaces and g be a linear operator from X into Y. Then g is bounded if and only if PreNorms(g) is upper bounded.

(32) Let X, Y be complex normed spaces. Then there exists a function N_{1} from BdLinOps(X, Y) into ℝ such that for every set f if f ∈ BdLinOps(X, Y), then N_{1}(f) = supPreNorms(modetrans(f, X, Y)).

Let X, Y be complex normed spaces. The functor BdLinOpsNorm(X, Y) yields a function from BdLinOps(X, Y) into ℝ and is defined by:

(Def. 12) For every set x such that x ∈ BdLinOps(X, Y) holds

(BdLinOpsNorm(X, Y))(x) = supPreNorms(modetrans(x, X, Y)). We now state two propositions:

(33) For all complex normed spaces X, Y and for every bounded linear operator f from X into Y holds modetrans(f, X, Y) = f.

(34) For all complex normed spaces X, Y and for every bounded linear operator f from X into Y holds (BdLinOpsNorm(X, Y))(f) = supPreNorms(f).

Let X, Y be complex normed spaces. The functor CNSpBdLinOps(X, Y) yields a non empty complex normed space structure and is defined by:

(Def. 13) CNSpBdLinOps(X, Y) = ⟨BdLinOps(X, Y), Zero (BdLinOps(X, Y), CVSpLinOps(X, Y)), Add (BdLinOps(X, Y), CVSpLinOps(X, Y)), Mult (BdLinOps(X, Y), CVSpLinOps(X, Y)), BdLinOpsNorm(X, Y)⟩.

The following four propositions are true:

(35) For all complex normed spaces X, Y holds (the carrier of X) ⟼→ 0_{Y} = 0_{CNSpBdLinOps (X,Y)}.

(36) Let X, Y be complex normed spaces, f be a point of CNSpBdLinOps(X, Y), and g be a bounded linear operator from X into Y. If g = f, then for every vector t of X holds ∥g(t)∥ ∥f∥ · ∥t∥.

(37) For all complex normed spaces X, Y and for every point f of CNSpBdLinOps(X, Y) holds 0 ∥f∥.

(38) For all complex normed spaces X, Y and for every point f of

CNSpBdLinOps(X, Y) such that f = 0_{CNSpBdLinOps (X,Y)} holds 0 = ∥f∥. Let X, Y be complex normed spaces. One can check that every element of

CNSpBdLinOps(X, Y) is function-like and relation-like.

Let X, Y be complex normed spaces, let f be an element of CNSpBdLinOps(X, Y), and let v be a vector of X. Then f(v) is a vector of Y.

We now state several propositions:

(39) Let X, Y be complex normed spaces and f, g, h be points of CNSpBdLinOps(X, Y). Then h = f + g if and only if for every vector x of X holds h(x) = f(x) + g(x).

(40) Let X, Y be complex normed spaces, f, h be points of CNSpBdLinOps(X, Y), and c be a Complex. Then h = c · f if and only if for every vector x of X holds h(x) = c · f(x).

(41) Let X, Y be complex normed spaces, f, g be points of CNSpBdLinOps(X, Y), and c be a Complex. Then ∥f∥ = 0 iff f = 0_{CNSpBdLinOps (X,Y)} and ∥c · f∥ = |c| · ∥f∥ and ∥f + g∥ ∥f∥ + ∥g∥.

(42) For all complex normed spaces X, Y holds CNSpBdLinOps(X, Y) is complex normed space-like.

(43) For all complex normed spaces X, Y holds CNSpBdLinOps(X, Y) is a

complex normed space. Let X, Y be complex normed spaces. Observe that CNSpBdLinOps(X, Y) is

complex normed space-like, complex linear space-like, Abelian, add-associative, right zeroed, and right complementable.

One can prove the following proposition

(44) Let X, Y be complex normed spaces and f, g, h be points of CNSpBdLinOps(X, Y). Then h = f − g if and only if for every vector x of X holds h(x) = f(x) − g(x).

4. Complex Banach Space of Bounded Linear Operators

Let X be a complex normed space. We say that X is complete if and only if:

(Def. 14) For every sequence s_{1} of X such that s_{1} is Cauchy sequence by norm holds s_{1} is convergent.

Let us observe that there exists a complex normed space which is complete. A complex Banach space is a complete complex normed space. One can prove the following three propositions:

(45) Let X be a complex normed space and s_{1} be a sequence of X. If s_{1} is convergent, then ∥s_{1}∥ is convergent and lim∥s_{1}∥ = ∥lims_{1}∥.

(46) Let X, Y be complex normed spaces. Suppose Y is complete. Let s_{1} be a sequence of CNSpBdLinOps(X, Y). If s_{1} is Cauchy sequence by norm, then s_{1} is convergent.

(47) For every complex normed space X and for every complex Banach space Y holds CNSpBdLinOps(X, Y) is a complex Banach space.

Let X be a complex normed space and let Y be a complex Banach space. One can verify that CNSpBdLinOps(X, Y) is complete.

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Complex Banach Space of Bounded Complex Sequences

Noboru Endou Gifu National College of Technology

Summary. An extension of [18]. In this article, we introduce two complex Banach spaces. One of them is the space of bounded complex sequences. The other one is the space of complex bounded functions, which is defined by the set of all complex bounded functions.

MML Identifier: CSSPACE4.

The articles [21], [6], [23], [24], [17], [20], [2], [19], [12], [4], [5], [7], [22], [3], [1], [16], [15], [14], [10], [13], [11], [8], and [9] provide the terminology and notation for this paper.

1. Complex Banach Space of Bounded Complex Sequences

The subset the set of bounded complex sequences of the linear space of complex sequences is defined by the condition (Def. 1).

(Def. 1) Let x be a set. Then x ∈ the set of bounded complex sequences if and

only if x ∈ the set of complex sequences and id_{seq}(x) is bounded. Let us note that the set of bounded complex sequences is non empty and

the set of bounded complex sequences is linearly closed.

One can prove the following proposition

(1) ⟨the set of bounded complex sequences, Zero (the set of bounded complex sequences, the linear space of complex sequences), Add (the set of bounded complex sequences, the linear space of complex sequences), Mult (the set of bounded complex sequences, the linear space of complex sequences)⟩ is a subspace of the linear space of complex sequences.

Let us mention that ⟨the set of bounded complex sequences, Zero (the set of bounded complex sequences, the linear space of complex sequences), Add (the set of bounded complex sequences, the linear space of complex sequences),

Mult (the set of bounded complex sequences, the linear space of complex sequences)⟩ is Abelian, add-associative, right zeroed, right complementable, and complex linear space-like.

The function Clinfty-norm from the set of bounded complex sequences into ℝ is defined by:

(Def. 2) For every set x such that x ∈ the set of bounded complex sequences

holds Clinfty-norm(x) = suprng| id_{seq}(x)|. Next we state the proposition

(2) For every complex sequence s_{1} holds s_{1} is bounded and suprng|s_{1}| = 0 iff for every natural number n holds s_{1}(n) = 0_{ℂ}.

One can check that ⟨the set of bounded complex sequences, Zero (the set of bounded complex sequences, the linear space of complex sequences), Add (the set of bounded complex sequences, the linear space of complex sequences),

Mult (the set of bounded complex sequences, the linear space of complex sequences), Clinfty-norm⟩ is Abelian, add-associative, right zeroed, right complementable, and complex linear space-like.

The non empty complex normed space structure Clinfty-Space is defined by the condition (Def. 3).

(Def. 3) Clinfty-Space = ⟨the set of bounded complex sequences, Zero (the set of bounded complex sequences, the linear space of complex sequences), Add (the set of bounded complex sequences, the linear space of complex sequences), Mult (the set of bounded complex sequences, the linear space of complex sequences), Clinfty-norm⟩.

Next we state two propositions:

(3) The carrier of Clinfty-Space = the set of bounded complex sequences and for every set x holds x is a vector of Clinfty-Space iff x is a complex sequence and id_{seq}(x) is bounded and 0_{Clinfty-Space} = CZeroseq and for every vector u of Clinfty-Space holds u = id_{seq}(u) and for all vectors u, v of Clinfty-Space holds u + v = id_{seq}(u) + id_{seq}(v) and for every Complex c and for every vector u of Clinfty-Space holds c·u = c id_{seq}(u) and for every vector u of Clinfty-Space holds − u = − id_{seq}(u) and id_{seq}(− u) = − id_{seq}(u) and for all vectors u, v of Clinfty-Space holds u − v = id_{seq}(u) − id_{seq}(v) and for every vector v of Clinfty-Space holds id_{seq}(v) is bounded and for every vector v of Clinfty-Space holds ∥v∥ = suprng| id_{seq}(v)|.

(4) Let x, y be points of Clinfty-Space and c be a Complex. Then ∥x∥ = 0 iff x = 0_{Clinfty-Space} and 0 ∥x∥ and ∥x+y∥ ∥x∥+∥y∥ and ∥c·x∥ = |c|·∥x∥.

Let us note that Clinfty-Space is complex normed space-like, complex linear space-like, Abelian, add-associative, right zeroed, and right complementable.

We now state two propositions:

(5) For every sequence v_{1} of Clinfty-Space such that v_{1} is Cauchy sequence by norm holds v_{1} is convergent.

(6) Clinfty-Space is a complex Banach space.

2. Another Example of Complex Banach Space

Let X be a non empty set, let Y be a complex normed space, and let I_{1} be a function from X into the carrier of Y. We say that I_{1} is bounded if and only if:

(Def. 4) There exists a real number K such that 0 K and for every element x

of X holds ∥I_{1}(x)∥ K. The following proposition is true

(7) Let X be a non empty set, Y be a complex normed space, and f be a function from X into the carrier of Y. If for every element x of X holds f(x) = 0_{Y} , then f is bounded.

Let X be a non empty set and let Y be a complex normed space. One can check that there exists a function from X into the carrier of Y which is bounded.

Let X be a non empty set and let Y be a complex normed space. The functor CBdFuncs(X, Y) yields a subset of ComplexVectSpace(X, Y) and is defined by:

(Def. 5) For every set x holds x ∈ CBdFuncs(X, Y) iff x is a bounded function

from X into the carrier of Y. Let X be a non empty set and let Y be a complex normed space. Note that

CBdFuncs(X, Y) is non empty.

One can prove the following propositions:

(8) For every non empty set X and for every complex normed space Y holds CBdFuncs(X, Y) is linearly closed.

(9) Let X be a non empty set and Y be a complex normed space. Then ⟨CBdFuncs(X, Y), Zero (CBdFuncs(X, Y), ComplexVectSpace(X, Y)), Add (CBdFuncs(X, Y), ComplexVectSpace(X, Y)), Mult (CBdFuncs(X, Y), ComplexVectSpace(X, Y))⟩ is a subspace of ComplexVectSpace(X, Y).

Let X be a non empty set and let Y be a complex normed space. Note that ⟨CBdFuncs(X, Y), Zero (CBdFuncs(X, Y), ComplexVectSpace(X, Y)),

Add (CBdFuncs(X, Y), ComplexVectSpace(X, Y)), Mult (CBdFuncs(X, Y), ComplexVectSpace(X, Y))⟩ is Abelian, add-associative, right zeroed, right

complementable, and complex linear space-like.

We now state the proposition (10) Let X be a non empty set and Y be a complex normed space. Then

⟨CBdFuncs(X, Y), Zero (CBdFuncs(X, Y), ComplexVectSpace(X, Y)), Add (CBdFuncs(X, Y), ComplexVectSpace(X, Y)), Mult (CBdFuncs(X, Y), ComplexVectSpace(X, Y))⟩ is a complex linear space.

Let X be a non empty set and let Y be a complex normed space. The set of bounded complex sequences from X into Y yielding a complex linear space is defined by:

(Def. 6) The set of bounded complex sequences from X into Y = ⟨CBdFuncs(X, Y), Zero (CBdFuncs(X, Y), ComplexVectSpace(X, Y)), Add (CBdFuncs(X, Y), ComplexVectSpace(X, Y)), Mult (CBdFuncs(X, Y), ComplexVectSpace(X, Y))⟩.

Let X be a non empty set and let Y be a complex normed space. One can verify that the set of bounded complex sequences from X into Y is strict.

The following three propositions are true:

(11) Let X be a non empty set, Y be a complex normed space, f, g, h be vectors of the set of bounded complex sequences from X into Y, and f′, g′, h′ be bounded functions from X into the carrier of Y. Suppose f′= f and g′= g and h′= h. Then h = f + g if and only if for every element x of X holds h′(x) = f′(x) + g′(x).

(12) Let X be a non empty set, Y be a complex normed space, f, h be vectors of the set of bounded complex sequences from X into Y, and f′, h′be bounded functions from X into the carrier of Y. Suppose f′= f and h′= h. Let c be a Complex. Then h = c · f if and only if for every element x of X holds h′(x) = c · f′(x).

(13) Let X be a non empty set and Y be a complex normed space. Then

0_{the set of bounded complex sequences from X into Y} = X ⟼→ 0_{Y} . Let X be a non empty set, let Y be a complex normed space, and let f be a

set. Let us assume that f ∈ CBdFuncs(X, Y). The functor modetrans(f, X, Y) yields a bounded function from X into the carrier of Y and is defined by:

(Def. 7) modetrans(f, X, Y) = f.

Let X be a non empty set, let Y be a complex normed space, and let u be a function from X into the carrier of Y. The functor PreNorms(u) yielding a non empty subset of ℝ is defined by:

(Def. 8) PreNorms(u) = {∥u(t)∥ : t ranges over elements of X}.

We now state three propositions:

(14) Let X be a non empty set, Y be a complex normed space, and g be a bounded function from X into the carrier of Y. Then PreNorms(g) is non empty and upper bounded.

(15) Let X be a non empty set, Y be a complex normed space, and g be a function from X into the carrier of Y. Then g is bounded if and only if

PreNorms(g) is upper bounded.

(16) Let X be a non empty set and Y be a complex normed space. Then there exists a function N_{1} from CBdFuncs(X, Y) into ℝ such that for every set f if f ∈ CBdFuncs(X, Y), then N_{1}(f) = supPreNorms(modetrans(f, X, Y)).

Let X be a non empty set and let Y be a complex normed space. The functor CBdFuncsNorm(X, Y) yielding a function from CBdFuncs(X, Y) into ℝ is defined by:

(Def. 9) For every set x such that x ∈ CBdFuncs(X, Y) holds

CBdFuncsNorm(X, Y)(x) = supPreNorms(modetrans(x, X, Y)). One can prove the following propositions:

(17) Let X be a non empty set, Y be a complex normed space, and f be a bounded function from X into the carrier of Y. Then modetrans(f, X, Y) = f.

(18) Let X be a non empty set, Y be a complex normed space, and f be a bounded function from X into the carrier of Y. Then CBdFuncsNorm(X, Y)(f) = supPreNorms(f).

Let X be a non empty set and let Y be a complex normed space. The complex normed space of bounded functions from X into Y yields a non empty complex normed space structure and is defined by:

(Def. 10) The complex normed space of bounded functions from X into Y = ⟨CBdFuncs(X, Y), Zero (CBdFuncs(X, Y), ComplexVectSpace(X, Y)), Add (CBdFuncs(X, Y), ComplexVectSpace(X, Y)), Mult (CBdFuncs(X, Y), ComplexVectSpace(X, Y)), CBdFuncsNorm(X, Y)⟩.

The following propositions are true:

(19) Let X be a non empty set and Y be a complex normed space. Then X ⟼→ 0_{Y} = 0_{the complex normed space of bounded functions from X into Y} .

(20) Let X be a non empty set, Y be a complex normed space, f be a point of the complex normed space of bounded functions from X into Y, and g be a bounded function from X into the carrier of Y. If g = f, then for every element t of X holds ∥g(t)∥ ∥f∥.

(21) Let X be a non empty set, Y be a complex normed space, and f be a point of the complex normed space of bounded functions from X into Y. Then 0 ∥f∥.

(22) Let X be a non empty set, Y be a complex normed space, and f be a point of the complex normed space of bounded functions from X into Y. Suppose f = 0_{the complex normed space of bounded functions from X into Y} . Then 0 = ∥f∥.

(23) Let X be a non empty set, Y be a complex normed space, f, g, h be points of the complex normed space of bounded functions from X into Y, and f′, g′, h′be bounded functions from X into the carrier of Y. Suppose f′ = f and g′ = g and h′ = h. Then h = f + g if and only if for every element x of X holds h′(x) = f′(x) + g′(x).

(24) Let X be a non empty set, Y be a complex normed space, f, h be points of the complex normed space of bounded functions from X into Y, and f′, h′ be bounded functions from X into the carrier of Y. Suppose f′= f and h′ = h. Let c be a Complex. Then h = c · f if and only if for every element x of X holds h′(x) = c · f′(x).

(25) Let X be a non empty set, Y be a complex normed space, f, g be points of the complex normed space of bounded functions from X into Y, and c be a Complex. Then

(i) ∥f∥ = 0 iff f = 0_{the complex normed space of bounded functions from X into Y} , (ii) ∥c · f∥ = |c| · ∥f∥, and (iii) ∥f + g∥ ∥f∥ + ∥g∥.

(26) Let X be a non empty set and Y be a complex normed space. Then the complex normed space of bounded functions from X into Y is complex normed space-like.

(27) Let X be a non empty set and Y be a complex normed space. Then the complex normed space of bounded functions from X into Y is a complex normed space.

Let X be a non empty set and let Y be a complex normed space. One can check that the complex normed space of bounded functions from X into Y is complex normed space-like, complex linear space-like, Abelian, add-associative, right zeroed, and right complementable.

One can prove the following three propositions:

(28) Let X be a non empty set, Y be a complex normed space, f, g, h be points of the complex normed space of bounded functions from X into Y, and f′, g′, h′be bounded functions from X into the carrier of Y. Suppose f′ = f and g′ = g and h′ = h. Then h = f − g if and only if for every element x of X holds h′(x) = f′(x) − g′(x).

(29) Let X be a non empty set and Y be a complex normed space. Suppose Y is complete. Let s_{1} be a sequence of the complex normed space of bounded functions from X into Y. If s_{1} is Cauchy sequence by norm, then s_{1} is convergent.

(30) Let X be a non empty set and Y be a complex Banach space. Then the complex normed space of bounded functions from X into Y is a complex Banach space.

Let X be a non empty set and let Y be a complex Banach space. Note that the complex normed space of bounded functions from X into Y is complete. complex banach space of bounded complex … 217

3. Some Properties of Complex Sequences

We now state four propositions:

(31) For all complex sequences s_{2}, s_{3} such that s_{2} is bounded and s_{3} is bounded holds s_{2} + s_{3} is bounded.

(32) For every Complex c and for every complex sequence s_{1} such that s_{1} is bounded holds c s_{1} is bounded.

(33) For every complex sequence s_{1} holds s_{1} is bounded iff |s_{1}| is bounded.

(34) For all complex sequences s_{2}, s_{3}, s_{4} holds s_{2} = s_{3}−s_{4} iff for every natural number n holds s_{2}(n) = s_{3}(n) − s_{4}(n).

References

[1] Agnieszka Banachowicz and Anna Winnicka. Complex sequences. Formalized Mathematics, 4(1):121–124, 1993.

[2] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91–96, 1990.

[3] Czesław Byliński. The complex numbers. Formalized Mathematics, 1(3):507–513, 1990.

[4] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55– 65, 1990.

[5] Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153–164, 1990.

[6] Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47–53, 1990.

[7] Czesław Byliński and Piotr Rudnicki. Bounding boxes for compact sets in E^{2}. Formalized Mathematics, 6(3):427–440, 1997.

[8] Noboru Endou. Banach space of absolute summable complex sequences. Formalized Mathematics, 12(2):191–194, 2004.

[9] Noboru Endou. Complex Banach space of bounded linear operators. Formalized Mathematics, 12(2):201–209, 2004.

[10] Noboru Endou. Complex linear space and complex normed space. Formalized Mathematics, 12(2):93–102, 2004.

[11] Noboru Endou. Complex linear space of complex sequences. Formalized Mathematics, 12(2):109–117, 2004.

[12] Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35–40, 1990.

[13] Jarosław Kotowicz. Convergent real sequences. Upper and lower bound of sets of real numbers. Formalized Mathematics, 1(3):477–481, 1990.

[14] Jarosław Kotowicz. Convergent sequences and the limit of sequences. Formalized Mathematics, 1(2):273–275, 1990.

[15] Jarosław Kotowicz. Real sequences and basic operations on them. Formalized Mathematics, 1(2):269–272, 1990.

[16] Adam Naumowicz. Conjugate sequences, bounded complex sequences and convergent complex sequences. Formalized Mathematics, 6(2):265–268, 1997.

[17] Beata Padlewska and Agata Darmochwał. Topological spaces and continuous functions. Formalized Mathematics, 1(1):223–230, 1990.

[18] Yasumasa Suzuki. Banach space of bounded real sequences. Formalized Mathematics, 12(2):77–83, 2004.

[19] Andrzej Trybulec. Subsets of complex numbers. To appear in Formalized Mathematics.

[20] Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics, 1(2):329–334, 1990.

[21] Andrzej Trybulec. Tarski Grothendieck set theory. Formalized Mathematics, 1(1):9–11, 1990.

[22] Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291–296,

218 noboru endou

[23] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67–71, 1990.

[24] Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181–186,

1990.

Received March 18, 2004

Concatenation of Finite Sequences Reducing Overlapping Part and an Argument

of Separators of Sequential Files

Hirofumi Fukura Shinshu University Nagano Yatsuka Nakamura Shinshu University Nagano

Summary. For two finite sequences, we present a notion of their concatenation, reducing overlapping part of the tail of the former and the head of the latter. At the same time, we also give a notion of common part of two finite sequences, which relates to the concatenation given here. A finite sequence is separated by another finite sequence (separator). We examined the condition that a separator separates uniquely any finite sequence. This will become a model of a separator of sequential files.

MML Identifier: FINSEQ 8.

The terminology and notation used here are introduced in the following articles: [14], [15], [9], [1], [12], [16], [3], [10], [2], [4], [5], [8], [13], [7], [11], and [6].

The following propositions are true:

(1) For every set D and for every finite sequence f of elements of D holds f↾0 = ∅.

(2) For every set D and for every finite sequence f of elements of D holds f_{⇂0} = f.

Let D be a set and let f, g be finite sequences of elements of D. Then f ^{⌢} g is a finite sequence of elements of D.

Next we state three propositions:

(3) For every non empty set D and for all finite sequences f, g of elements of D such that lenf 1 holds mid(f ^{⌢} g, 1, lenf) = f.

(4) Let D be a set, f be a finite sequence of elements of D, and i be a natural number. If i lenf, then f_{⇂i} = ε_{D}.

(5) For every non empty set D and for all natural numbers k_{1}, k_{2} holds mid(ε_{D}, k_{1}, k_{2}) = ε_{D}.

Let D be a set, let f be a finite sequence of elements of D, and let k_{1}, k_{2} be natural numbers. The functor smid(f, k_{1}, k_{2}) yields a finite sequence of elements of D and is defined as follows:

(Def. 1) smid(f, k_{1}, k_{2}) = f_{⇂k_{1}− ′1}↾((k_{2} + 1) − ′k_{1}). One can prove the following propositions:

(6) Let D be a non empty set, f be a finite sequence of elements of D, and k_{1}, k_{2} be natural numbers. If k_{1} k_{2}, then smid(f, k_{1}, k_{2}) = mid(f, k_{1}, k_{2}).

(7) Let D be a non empty set, f be a finite sequence of elements of D, and k_{2} be a natural number. Then smid(f, 1, k_{2}) = f↾k_{2}.

(8) Let D be a non empty set, f be a finite sequence of elements of D, and k_{2} be a natural number. If lenf k_{2}, then smid(f, 1, k_{2}) = f.

(9) Let D be a set, f be a finite sequence of elements of D, and k_{1}, k_{2} be natural numbers. If k_{1} > k_{2}, then smid(f, k_{1}, k_{2}) = ∅ and smid(f, k_{1}, k_{2}) = ε_{D}.

(10) For every set D and for every finite sequence f of elements of D and for every natural number k_{2} holds smid(f, 0, k_{2}) = smid(f, 1, k_{2} + 1).

(11) For every non empty set D and for all finite sequences f, g of elements

of D holds smid(f ^{⌢} g, lenf + 1, lenf + leng) = g. Let D be a non empty set and let f, g be finite sequences of elements of D.

The functor ovlpart(f, g) yielding a finite sequence of elements of D is defined by the conditions (Def. 2).

(Def. 2)(i) lenovlpart(f, g) leng, (ii) ovlpart(f, g) = smid(g, 1, lenovlpart(f, g)), (iii) ovlpart(f, g) = smid(f, (lenf − ′lenovlpart(f, g)) + 1, lenf), and (iv) for every natural number j such that j leng and smid(g, 1, j) =

smid(f, (lenf − ′j) + 1, lenf) holds j lenovlpart(f, g). Next we state the proposition

(12) For every non empty set D and for all finite sequences f, g of elements

of D holds lenovlpart(f, g) lenf. Let D be a non empty set and let f, g be finite sequences of elements of D.

The functor ovlcon(f, g) yielding a finite sequence of elements of D is defined as follows:

(Def. 3) ovlcon(f, g) = f ^{⌢} (g_{⇂len ovlpart(f,g)}).

One can prove the following proposition

(13) For every non empty set D and for all finite sequences f, g of elements of D holds ovlcon(f, g) = (f↾(lenf − ′lenovlpart(f, g)))^{⌢} g.

Let D be a non empty set and let f, g be finite sequences of elements of D. The functor ovlldiff(f, g) yields a finite sequence of elements of D and is defined as follows:

(Def. 4) ovlldiff(f, g) = f↾(lenf − ′lenovlpart(f, g)).

Let D be a non empty set and let f, g be finite sequences of elements of D. The functor ovlrdiff(f, g) yields a finite sequence of elements of D and is defined by:

(Def. 5) ovlrdiff(f, g) = g_{⇂len ovlpart(f,g)}.

One can prove the following propositions:

(14) Let D be a non empty set and f, g be finite sequences of elements of D. Then ovlcon(f, g) = (ovlldiff(f, g))^{⌢} ovlpart(f, g)^{⌢} ovlrdiff(f, g) and ovlcon(f, g) = (ovlldiff(f, g))^{⌢} ((ovlpart(f, g))^{⌢} ovlrdiff(f, g)).

(15) Let D be a non empty set and f be a finite sequence of elements of D. Then ovlcon(f, f) = f and ovlpart(f, f) = f and ovlldiff(f, f) = ∅ and ovlrdiff(f, f) = ∅.

(16) For every non empty set D and for all finite sequences f, g of elements of D holds ovlpart(f ^{⌢} g, g) = g and ovlpart(f, f ^{⌢} g) = f.

(17) Let D be a non empty set and f, g be finite sequences of elements of D. Then lenovlcon(f, g) = (lenf + leng) − lenovlpart(f, g) and lenovlcon(f, g) = (lenf + leng) − ′lenovlpart(f, g) and lenovlcon(f, g) = lenf + (leng − ′lenovlpart(f, g)).

(18) For every non empty set D and for all finite sequences f, g of elements of D holds lenovlpart(f, g) lenf and lenovlpart(f, g) leng.

Let D be a non empty set and let C_{1} be a finite sequence of elements of D. We say that C_{1} separates uniquely if and only if the condition (Def. 6) is satisfied.

(Def. 6) Let f be a finite sequence of elements of D and i, j be natural numbers. Suppose 1 i and i < j and (j + lenC_{1}) − ′1 lenf and smid(f, i, (i + lenC_{1})− ′1) = smid(f, j, (j +lenC_{1})− ′1) and smid(f, i, (i+lenC_{1})− ′1) = C_{1}. Then j − ′i lenC_{1}.

The following proposition is true

(19) Let D be a non empty set and C_{1} be a finite sequence of elements of D. Then C_{1} separates uniquely if and only if lenovlpart((C_{1})_{⇂1}, C_{1}) = 0.

Let D be a non empty set, let f, g be finite sequences of elements of D, and let n be a natural number. We say that g is a substring of f if and only if:

(Def. 7) If leng > 0, then there exists a natural number i such that n i and

i lenf and mid(f, i, (i − ′1) + leng) = g. We now state four propositions:

(20) Let D be a non empty set, f, g be finite sequences of elements of D, and n be a natural number. If leng = 0, then g is a substring of f.

(21) Let D be a non empty set, f, g be finite sequences of elements of D, and n, m be natural numbers. If m n and g is a substring of f, then g is a substring of f.

(22) For every non empty set D and for every finite sequence f of elements of D such that 1 lenf holds f is a substring of f.

(23) Let D be a non empty set and f, g be finite sequences of elements of D.

If g is a substring of f, then g is a substring of f. Let D be a non empty set and let f, g be finite sequences of elements of D.

We say that g is a preposition of f if and only if: (Def. 8) If leng > 0, then 1 lenf and mid(f, 1, leng) = g.

One can prove the following four propositions:

(24) Let D be a non empty set and f, g be finite sequences of elements of D. If leng = 0, then g is a preposition of f.

(25) For every non empty set D holds every finite sequence f of elements of D is a preposition of f.

(26) Let D be a non empty set and f, g be finite sequences of elements of D. If g is a preposition of f, then leng lenf.

(27) Let D be a non empty set and f, g be finite sequences of elements of D.

If leng > 0 and g is a preposition of f, then g(1) = f(1). Let D be a non empty set and let f, g be finite sequences of elements of D.

We say that g is a postposition of f if and only if: (Def. 9) Rev(g) is a preposition of Rev(f).

Next we state several propositions:

(28) Let D be a non empty set and f, g be finite sequences of elements of D. If leng = 0, then g is a postposition of f.

(29) Let D be a non empty set and f, g be finite sequences of elements of D. If g is a postposition of f, then leng lenf.

(30) Let D be a non empty set, f, g be finite sequences of elements of D, and n be a natural number. Suppose g is a postposition of f. If leng > 0, then leng lenf and mid(f, (lenf + 1) − ′leng, lenf) = g.

(31) Let D be a non empty set, f, g be finite sequences of elements of D, and n be a natural number such that if leng > 0, then leng lenf and mid(f, (lenf + 1) − ′leng, lenf) = g. Then g is a postposition of f.

(32) Let D be a non empty set, f, g be finite sequences of elements of D, and n be a natural number. If leng = 0, then g is a preposition of f.

(33) Let D be a non empty set, f, g be finite sequences of elements of D, and n be a natural number. If 1 lenf and g is a preposition of f, then g is

concatenation of finite sequences reducing … 223

a substring of f.

(34) Let D be a non empty set, f, g be finite sequences of elements of D, and n be a natural number. Suppose g is not a substring of f. Let i be a natural number. If n i and 0 < i, then mid(f, i, (i − ′1) + leng) ≠ g.

Let D be a non empty set, let f, g be finite sequences of elements of D, and let n be a natural number. The functor instr(n, f) yielding a natural number is defined by the conditions (Def. 10).

(Def. 10)(i) If instr(n, f) ≠ 0, then n instr(n, f) and g is a preposition of f_{⇂instr(n,f)− ′1} and for every natural number j such that j n and j > 0 and g is a preposition of f_{⇂j− ′1} holds j instr(n, f), and

(ii) if instr(n, f) = 0, then g is not a substring of f. Let D be a non empty set and let f, C_{1} be finite sequences of elements of D.

The functor addcr(f, C_{1}) yields a finite sequence of elements of D and is defined by:

(Def. 11) addcr(f, C_{1}) = ovlcon(f, C_{1}).

Let D be a non empty set and let r, C_{1} be finite sequences of elements of D. We say that r is terminated by C_{1} if and only if:

(Def. 12) If lenC_{1} > 0, then lenr lenC_{1} and instr(1, r) = (lenr + 1) − ′lenC_{1}.

The following proposition is true

(35) For every non empty set D holds every finite sequence f of elements of D is terminated by f.

References

[1] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41–46, 1990.

[2] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107–114, 1990.

[3] Czesław Byliński. Finite sequences and tuples of elements of a non-empty sets. Formalized Mathematics, 1(3):529–536, 1990.

[4] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55– 65, 1990.

[5] Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153–164, 1990.

[6] Czesław Byliński. Some properties of restrictions of finite sequences. Formalized Mathematics, 5(2):241–245, 1996.

[7] Agata Darmochwał. The Euclidean space. Formalized Mathematics, 2(4):599–603, 1991.

[8] Agata Darmochwał and Yatsuka Nakamura. The topological space E_{T}^{2}. Arcs, line segments and special polygonal arcs. Formalized Mathematics, 2(5):617–621, 1991.

[9] Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35–40, 1990.

[10] Jarosław Kotowicz. Functions and finite sequences of real numbers. Formalized Mathematics, 3(2):275–278, 1992.

[11] Yatsuka Nakamura and Roman Matuszewski. Reconstructions of special sequences. Formalized Mathematics, 6(2):255–263, 1997.

[12] Takaya Nishiyama and Yasuho Mizuhara. Binary arithmetics. Formalized Mathematics, 4(1):83–86, 1993.

224 hirofumi fukura and yatsuka nakamura

[13] Beata Padlewska and Agata Darmochwał. Topological spaces and continuous functions. Formalized Mathematics, 1(1):223–230, 1990.

[14] Andrzej Trybulec. Tarski Grothendieck set theory. Formalized Mathematics, 1(1):9–11, 1990.

[15] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67–71, 1990.

[16] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73–83, 1990.

Received March 18, 2004

Cauchy Sequence of Complex Unitary Space

Yasumasa Suzuki Take, Yokosuka-shi Japan

Noboru Endou Gifu National College of Technology

Summary. As an extension of [13], we introduce the Cauchy sequence of complex unitary space and describe its properties.

MML Identifier: CLVECT 3.

The terminology and notation used in this paper are introduced in the following papers: [22], [3], [20], [9], [5], [12], [10], [11], [15], [2], [18], [4], [1], [21], [16], [17], [14], [13], [19], [6], [7], and [8].

For simplicity, we follow the rules: X denotes a complex unitary space, s_{1}, s_{2}, s_{3} denote sequences of X, R_{1} denotes a sequence of real numbers, C_{1}, C_{2}, C_{3} denote complex sequences, z, z_{1}, z_{2} denote Complexes, r denotes a real number, and k, n, m denote natural numbers.

The scheme Rec Func Ex CUS deals with a complex unitary space A, a point B of A, and a binary functor F yielding a point of A, and states that:

There exists a function f from ℕ into the carrier of A such that f(0) = B and for every element n of ℕ and for every point x of A such that x = f(n) holds f(n + 1) = F(n, x)

for all values of the parameters.

Let us consider X, s_{1}. The functor (∑_{α=0}^{κ}(s_{1})(α))_{κ∈ℕ} yields a sequence of X and is defined as follows:

(Def. 1) (∑_{α=0}^{κ}(s_{1})(α))_{κ∈ℕ}(0) = s_{1}(0) and for every n holds (∑_{α=0}^{κ}(s_{1})(α))_{κ∈ℕ}(n+

1) = (∑_{α=0}^{κ}(s_{1})(α))_{κ∈ℕ}(n) + s_{1}(n + 1). One can prove the following propositions:

(1) (∑_{α=0}^{κ}(s_{2})(α))_{κ∈ℕ} + (∑_{α=0}^{κ}(s_{3})(α))_{κ∈ℕ} = (∑_{α=0}^{κ}(s_{2} + s_{3})(α))_{κ∈ℕ}.

(2) (∑_{α=0}^{κ}(s_{2})(α))_{κ∈ℕ} − (∑_{α=0}^{κ}(s_{3})(α))_{κ∈ℕ} = (∑_{α=0}^{κ}(s_{2} − s_{3})(α))_{κ∈ℕ}.

(3) (∑_{α=0}^{κ}(z · s_{1})(α))_{κ∈ℕ} = z · (∑_{α=0}^{κ}(s_{1})(α))_{κ∈ℕ}.

(4) (∑_{α=0}^{κ}(− s_{1})(α))_{κ∈ℕ} = − (∑_{α=0}^{κ}(s_{1})(α))_{κ∈ℕ}.

(5) z_{1} · (∑_{α=0}^{κ}(s_{2})(α))_{κ∈ℕ} + z_{2} · (∑_{α=0}^{κ}(s_{3})(α))_{κ∈ℕ} = (∑_{α=0}^{κ}(z_{1} · s_{2} + z_{2} · s_{3})(α))_{κ∈ℕ}.

Let us consider X, s_{1}. We say that s_{1} is summable if and only if: (Def. 2) (∑_{α=0}^{κ}(s_{1})(α))_{κ∈ℕ} is convergent.

The functor ∑ s_{1} yields a point of X and is defined as follows: (Def. 3) ∑ s_{1} = lim((∑_{α=0}^{κ}(s_{1})(α))_{κ∈ℕ}).

Next we state several propositions:

(6) If s_{2} is summable and s_{3} is summable, then s_{2} + s_{3} is summable and ∑(s_{2} + s_{3}) = ∑ s_{2} + ∑ s_{3}.

(7) If s_{2} is summable and s_{3} is summable, then s_{2} − s_{3} is summable and ∑(s_{2} − s_{3}) = ∑ s_{2} − ∑ s_{3}.

(8) If s_{1} is summable, then z · s_{1} is summable and ∑(z · s_{1}) = z · ∑ s_{1}.

(9) If s_{1} is summable, then s_{1} is convergent and lims_{1} = 0_{X}.

(10) Suppose X is Hilbert. Then s_{1} is summable if and only if for every r such that r > 0 there exists k such that for all n, m such that n k and m k holds ∥(∑_{α=0}^{κ}(s_{1})(α))_{κ∈ℕ}(n) − (∑_{α=0}^{κ}(s_{1})(α))_{κ∈ℕ}(m)∥ < r.

(11) If s_{1} is summable, then (∑_{α=0}^{κ}(s_{1})(α))_{κ∈ℕ} is bounded.

(12) If for every n holds s_{2}(n) = s_{1}(0), then (∑_{α=0}^{κ}(s_{1} ↑ 1)(α))_{κ∈ℕ} = (∑_{α=0}^{κ}(s_{1})(α))_{κ∈ℕ} ↑ 1 − s_{2}.

(13) If s_{1} is summable, then for every k holds s_{1} ↑ k is summable.

(14) If there exists k such that s_{1} ↑ k is summable, then s_{1} is summable. Let us consider X, s_{1}, n. The functor ∑_{κ=0}^{n} s_{1}(κ) yielding a point of X is

defined by: (Def. 4) ∑_{κ=0}^{n} s_{1}(κ) = (∑_{α=0}^{κ}(s_{1})(α))_{κ∈ℕ}(n).

One can prove the following propositions:

(15) ∑_{κ=0}^{0} s_{1}(κ) = s_{1}(0).

(16) ∑_{κ=0}^{1} s_{1}(κ) = ∑_{κ=0}^{0} s_{1}(κ) + s_{1}(1).

(17) ∑_{κ=0}^{1} s_{1}(κ) = s_{1}(0) + s_{1}(1).

(18) ∑_{κ=0}^{n+1} s_{1}(κ) = ∑_{κ=0}^{n} s_{1}(κ) + s_{1}(n + 1).

(19) s_{1}(n + 1) = ∑_{κ=0}^{n+1} s_{1}(κ) − ∑_{κ=0}^{n} s_{1}(κ).

(20) s_{1}(1) = ∑_{κ=0}^{1} s_{1}(κ) − ∑_{κ=0}^{0} s_{1}(κ).

Let us consider X, s_{1}, n, m. The functor ∑_{κ=n+1}^{m} s_{1}(κ) yielding a point of X is defined by:

(Def. 5) ∑_{κ=n+1}^{m} s_{1}(κ) = ∑_{κ=0}^{n} s_{1}(κ) − ∑_{κ=0}^{m} s_{1}(κ).

One can prove the following four propositions:

(21) ∑_{κ=1+1}^{0} s_{1}(κ) = s_{1}(1).

(22) ∑_{κ=n+1+1}^{n} s_{1}(κ) = s_{1}(n + 1).

(23) Suppose X is Hilbert. Then s_{1} is summable if and only if for every r such that r > 0 there exists k such that for all n, m such that n k and m k holds ∥∑_{κ=0}^{n} s_{1}(κ) − ∑_{κ=0}^{m} s_{1}(κ)∥ < r.

(24) Suppose X is Hilbert. Then s_{1} is summable if and only if for every r such that r > 0 there exists k such that for all n, m such that n k and m k holds ∥∑_{κ=n+1}^{m} s_{1}(κ)∥ < r.

Let us consider C_{1}, n. The functor ∑_{κ=0}^{n} C_{1}(κ) yielding a Complex is defined as follows:

(Def. 6) ∑_{κ=0}^{n} C_{1}(κ) = (∑_{α=0}^{κ}(C_{1})(α))_{κ∈ℕ}(n).

Let us consider C_{1}, n, m. The functor ∑_{κ=n+1}^{m} C_{1}(κ) yielding a Complex is defined by:

(Def. 7) ∑_{κ=n+1}^{m} C_{1}(κ) = ∑_{κ=0}^{n} C_{1}(κ) − ∑_{κ=0}^{m} C_{1}(κ).

Let us consider X, s_{1}. We say that s_{1} is absolutely summable if and only if: (Def. 8) ∥s_{1}∥ is summable.

The following propositions are true:

(25) If s_{2} is absolutely summable and s_{3} is absolutely summable, then s_{2} +s_{3} is absolutely summable.

(26) If s_{1} is absolutely summable, then z · s_{1} is absolutely summable.

(27) If for every n holds ∥s_{1}∥(n) R_{1}(n) and R_{1} is summable, then s_{1} is absolutely summable.

(28) If for every n holds s_{1}(n) ≠ 0_{X} and R_{1}(n) = \frac{∥s_{1}(n+1)∥}{∥s_{1}(n)∥} and R_{1} is convergent and limR_{1} < 1, then s_{1} is absolutely summable.

(29) If r > 0 and there exists m such that for every n such that n m holds ∥s_{1}(n)∥ r, then s_{1} is not convergent or lims_{1} ≠ 0_{X}.

(30) If for every n holds s_{1}(n) ≠ 0_{X} and there exists m such that for every n such that n m holds \frac{∥s_{1}(n+1)∥}{∥s_{1}(n)∥} 1, then s_{1} is not summable.

(31) If for every n holds s_{1}(n) ≠ 0_{X} and for every n holds R_{1}(n) = \frac{∥s_{1}(n+1)∥}{∥s_{1}(n)∥} and R_{1} is convergent and limR_{1} > 1, then s_{1} is not summable.

(32) If for every n holds R_{1}(n) = ^{n} \sqrt{∥s_{1}(n)∥} and R_{1} is convergent and limR_{1} < 1, then s_{1} is absolutely summable.

(33) If for every n holds R_{1}(n) = ^{n} \sqrt{∥s_{1}∥(n)} and there exists m such that for every n such that n m holds R_{1}(n) 1, then s_{1} is not summable.

(34) If for every n holds R_{1}(n) = ^{n} \sqrt{∥s_{1}∥(n)} and R_{1} is convergent and limR_{1} > 1, then s_{1} is not summable.

(35) (∑_{α=0}^{κ}∥s_{1}∥(α))_{κ∈ℕ} is non-decreasing.

(36) For every n holds (∑_{α=0}^{κ}∥s_{1}∥(α))_{κ∈ℕ}(n) 0.

(37) For every n holds ∥(∑_{α=0}^{κ}(s_{1})(α))_{κ∈ℕ}(n)∥ (∑_{α=0}^{κ}∥s_{1}∥(α))_{κ∈ℕ}(n).

(38) For every n holds ∥∑_{κ=0}^{n} s_{1}(κ)∥ ∑_{κ=0}^{n}∥s_{1}∥(κ).

(39) For all n, m holds ∥(∑_{α=0}^{κ}(s_{1})(α))_{κ∈ℕ}(m) − (∑_{α=0}^{κ}(s_{1})(α))_{κ∈ℕ}(n)∥ |(∑_{α=0}^{κ}∥s_{1}∥(α))_{κ∈ℕ}(m) − (∑_{α=0}^{κ}∥s_{1}∥(α))_{κ∈ℕ}(n)|.

(40) For all n, m holds ∥∑_{κ=0}^{m} s_{1}(κ) − ∑_{κ=0}^{n} s_{1}(κ)∥ |∑_{κ=0}^{m}∥s_{1}∥(κ) − ∑_{κ=0}^{n}∥s_{1}∥(κ)|.

(41) For all n, m holds ∥∑_{κ=m+1}^{n} s_{1}(κ)∥ |∑_{κ=m+1}^{n}∥s_{1}∥(κ)|.

(42) If X is Hilbert, then if s_{1} is absolutely summable, then s_{1} is summable. Let us consider X, s_{1}, C_{1}. The functor C_{1} · s_{1} yields a sequence of X and is

defined by: (Def. 9) For every n holds (C_{1} · s_{1})(n) = C_{1}(n) · s_{1}(n).

Next we state several propositions:

(43) C_{1} · (s_{2} + s_{3}) = C_{1} · s_{2} + C_{1} · s_{3}.

(44) (C_{2} + C_{3}) · s_{1} = C_{2} · s_{1} + C_{3} · s_{1}.

(45) (C_{2} C_{3}) · s_{1} = C_{2} · (C_{3} · s_{1}).

(46) (z C_{1}) · s_{1} = z · (C_{1} · s_{1}).

(47) C_{1} · − s_{1} = (− C_{1}) · s_{1}.

(48) If C_{1} is convergent and s_{1} is convergent, then C_{1} · s_{1} is convergent.

(49) If C_{1} is bounded and s_{1} is bounded, then C_{1} · s_{1} is bounded.

(50) If C_{1} is convergent and s_{1} is convergent, then C_{1} · s_{1} is convergent and

lim(C_{1} · s_{1}) = limC_{1} · lims_{1}. Let us consider C_{1}. We say that C_{1} is Cauchy if and only if:

(Def. 10) For every r such that r > 0 there exists k such that for all n, m such

that n k and m k holds |C_{1}(n) − C_{1}(m)| < r. We introduce C_{1} is a Cauchy sequence as a synonym of C_{1} is Cauchy.

Next we state four propositions:

(51) If X is Hilbert, then if s_{1} is Cauchy and C_{1} is Cauchy, then C_{1} · s_{1} is Cauchy.

(52) For every n holds (∑_{α=0}^{κ}((C_{1} − C_{1} ↑1) · (∑_{α=0}^{κ}(s_{1})(α))_{κ∈ℕ})(α))_{κ∈ℕ}(n) = (∑_{α=0}^{κ}(C_{1} · s_{1})(α))_{κ∈ℕ}(n + 1) − (C_{1} · (∑_{α=0}^{κ}(s_{1})(α))_{κ∈ℕ})(n + 1).

(53) For every n holds (∑_{α=0}^{κ}(C_{1} · s_{1})(α))_{κ∈ℕ}(n + 1) = (C_{1} · (∑_{α=0}^{κ}(s_{1})(α))_{κ∈ℕ})(n+1)−(∑_{α=0}^{κ}((C_{1}↑1−C_{1})·(∑_{α=0}^{κ}(s_{1})(α))_{κ∈ℕ})(α))_{κ∈ℕ}(n).

(54) For every n holds ∑_{κ=0}^{n+1}(C_{1} · s_{1})(κ) = (C_{1} · (∑_{α=0}^{κ}(s_{1})(α))_{κ∈ℕ})(n + 1) − ∑_{κ=0}^{n}((C_{1} ↑ 1 − C_{1}) · (∑_{α=0}^{κ}(s_{1})(α))_{κ∈ℕ})(κ).

References

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[2] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41–46, 1990.

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cauchy sequence of complex unitary space 229

[5] Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153–164, 1990.

[6] Noboru Endou. Complex linear space and complex normed space. Formalized Mathematics, 12(2):93–102, 2004.

[7] Noboru Endou. Complex linear space of complex sequences. Formalized Mathematics, 12(2):109–117, 2004.

[8] Noboru Endou. Convergent sequences in complex unitary space. Formalized Mathematics, 12(2):159–165, 2004.

[9] Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35–40, 1990.

[10] Jarosław Kotowicz. Convergent sequences and the limit of sequences. Formalized Mathematics, 1(2):273–275, 1990.

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[12] Jarosław Kotowicz. Real sequences and basic operations on them. Formalized Mathematics, 1(2):269–272, 1990.

[13] Elżbieta Kraszewska and Jan Popiołek. Series in Banach and Hilbert Spaces. Formalized Mathematics, 2(5):695–699, 1991.

[14] Adam Naumowicz. Conjugate sequences, bounded complex sequences and convergent complex sequences. Formalized Mathematics, 6(2):265–268, 1997.

[15] Jan Popiołek. Some properties of functions modul and signum. Formalized Mathematics, 1(2):263–264, 1990.

[16] Jan Popiołek. Real normed space. Formalized Mathematics, 2(1):111–115, 1991.

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Received March 18, 2004 230 yasumasa suzuki and noboru endou

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Index of MML Identifiers

AMISTD 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 CLOPBAN1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 CLVECT 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 CLVECT 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .159 CLVECT 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .225 CSSPACE2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 CSSPACE3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 CSSPACE4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 CSSPACE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 FINSEQ 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .219 FINTOPO3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 JORDAN20 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 LFUZZY 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 LOPBAN 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 LOPBAN 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .103 LOPBAN 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .131 LOPBAN 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .173 NAT 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 NECKLA 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .143 POLYEQ 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 PRGCOR 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 RADIX 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 RADIX 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 RECDEF 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .167 RFINSEQ2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1 ROUGHS 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 RSSPACE4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 SCMRING4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 SIN COS4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .139 TAYLOR 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .195 UNIROOTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 UPROOTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .49 WEDDWITT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

Contents Formaliz. Math. 12 (2)

Banach Space of Bounded Real Sequences

By Yasumasa Suzuki . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

Solving Roots of Polynomial Equation of Degree 2 and 3 with

Complex Coefficients

By Yuzhong Ding and Xiquan Liang  . . . . . . . . . . . . . . . . . . . . . . 85

Complex Linear Space and Complex Normed Space

By Noboru Endou  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

The Banach Algebra of Bounded Linear Operators

By Yasunari Shidama . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

Complex Linear Space of Complex Sequences

By Noboru Endou  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

Behaviour of an Arc Crossing a Line

By Yatsuka Nakamura  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

Some Set Series in Finite Topological Spaces. Fundamental Con-

cepts for Image Processing

By Masami Tanaka and Yatsuka Nakamura  . . . . . . . . . . . . . . 125

The Series on Banach Algebra

By Yasunari Shidama . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

Formulas and Identities of Trigonometric Functions

By Pacharapokin Chanapat et al.  . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

The Class of Series-Parallel Graphs. Part III

By Krzysztof Retel  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

Relocability for SCM over Ring

By Artur Korniłowicz and Yasunari Shidama  . . . . . . . . . . . 151

Convergent Sequences in Complex Unitary Space

By Noboru Endou  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

Continued on inside back cover

Recursive Definitions. Part II

By Artur Korniłowicz  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

The Exponential Function on Banach Algebra

By Yasunari Shidama . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

Fundamental Theorem of Arithmetic

By Artur Korniłowicz and Piotr Rudnicki  . . . . . . . . . . . . . . 179

Hilbert Space of Complex Sequences

By Noboru Endou  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

Banach Space of Absolute Summable Complex Sequences

By Noboru Endou  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

The Taylor Expansions

By Yasunari Shidama . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

Complex Banach Space of Bounded Linear Operators

By Noboru Endou  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

Complex Banach Space of Bounded Complex Sequences

By Noboru Endou  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

Concatenation of Finite Sequences Reducing Overlapping Part and

an Argument of Separators of Sequential Files

By Hirofumi Fukura and Yatsuka Nakamura  . . . . . . . . . . . . 219

Cauchy Sequence of Complex Unitary Space

By Yasumasa Suzuki and Noboru Endou  . . . . . . . . . . . . . . . . . . 225

Index of MML Identifiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231