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1
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Separability of Real Normed Spaces and Its Basic Properties

100%
Nakasho K., Endou N.
Formalized Mathematics
|
2015
|
tom 23
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nr 1
59-65
EN
In this article, the separability of real normed spaces and its properties are mainly formalized. In the first section, it is proved that a real normed subspace is separable if it is generated by a countable subset. We used here the fact that the rational numbers form a dense subset of the real numbers. In the second section, the basic properties of the separable normed spaces are discussed. It is applied to isomorphic spaces via bounded linear operators and double dual spaces. In the last section, it is proved that the completeness and reflexivity are transferred to sublinear normed spaces. The formalization is based on [34], and also referred to [7], [14] and [16].
2
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F. Riesz Theorem

81%
Narita K., Nakasho K., Shidama Y.
Formalized Mathematics
|
2017
|
tom 25
|
nr 3
179-184
EN
In this article, we formalize in the Mizar system [1, 4] the F. Riesz theorem. In the first section, we defined Mizar functor ClstoCmp, compact topological spaces as closed interval subset of real numbers. Then using the former definition and referring to the article [10] and the article [5], we defined the normed spaces of continuous functions on closed interval subset of real numbers, and defined the normed spaces of bounded functions on closed interval subset of real numbers. We also proved some related properties. In Sec.2, we proved some lemmas for the proof of F. Riesz theorem. In Sec.3, we proved F. Riesz theorem, about the dual space of the space of continuous functions on closed interval subset of real numbers, finally. We applied Hahn-Banach theorem (36) in [7], to the proof of the last theorem. For the description of theorems of this section, we also referred to the article [8] and the article [6]. These formalizations are based on [2], [3], [9], and [11].
3
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Compactness in Metric Spaces

81%
Nakasho K., Narita K., Shidama Y.
Formalized Mathematics
|
2016
|
tom 24
|
nr 3
167-172
EN
In this article, we mainly formalize in Mizar [2] the equivalence among a few compactness definitions of metric spaces, norm spaces, and the real line. In the first section, we formalized general topological properties of metric spaces. We discussed openness and closedness of subsets in metric spaces in terms of convergence of element sequences. In the second section, we firstly formalize the definition of sequentially compact, and then discuss the equivalence of compactness, countable compactness, sequential compactness, and totally boundedness with completeness in metric spaces. In the third section, we discuss compactness in norm spaces. We formalize the equivalence of compactness and sequential compactness in norm space. In the fourth section, we formalize topological properties of the real line in terms of convergence of real number sequences. In the last section, we formalize the equivalence of compactness and sequential compactness in the real line. These formalizations are based on [20], [5], [17], [14], and [4].
4
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Conservation Rules of Direct Sum Decomposition of Groups

81%
Nakasho K., Yamazaki H., Okazaki H., Shidama Y.
Formalized Mathematics
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2016
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tom 24
|
nr 1
81-94
EN
In this article, conservation rules of the direct sum decomposition of groups are mainly discussed. In the first section, we prepare miscellaneous definitions and theorems for further formalization in Mizar [5]. In the next three sections, we formalized the fact that the property of direct sum decomposition is preserved against the substitutions of the subscript set, flattening of direct sum, and layering of direct sum, respectively. We referred to [14], [13] [6] and [11] in the formalization.
5
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Rank of Submodule, Linear Transformations and Linearly Independent Subsets of Z-module

81%
Nakasho K., Futa Y., Okazaki H., Shidama Y.
Formalized Mathematics
|
2014
|
tom 22
|
nr 3
189-198
EN
In this article, we formalize some basic facts of Z-module. In the first section, we discuss the rank of submodule of Z-module and its properties. Especially, we formally prove that the rank of any Z-module is equal to or more than that of its submodules, and vice versa, and that there exists a submodule with any given rank that satisfies the above condition. In the next section, we mention basic facts of linear transformations between two Z-modules. In this section, we define homomorphism between two Z-modules and deal with kernel and image of homomorphism. In the last section, we formally prove some basic facts about linearly independent subsets and linear combinations. These formalizations are based on [9](p.191-242), [23](p.117-172) and [2](p.17-35).
6
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Topological Properties of Real Normed Space

81%
Nakasho K., Futa Y., Shidama Y.
Formalized Mathematics
|
2014
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tom 22
|
nr 3
209-223
EN
In this article, we formalize topological properties of real normed spaces. In the first part, open and closed, density, separability and sequence and its convergence are discussed. Then we argue properties of real normed subspace. Then we discuss linear functions between real normed speces. Several kinds of subspaces induced by linear functions such as kernel, image and inverse image are considered here. The fact that Lipschitz continuity operators preserve convergence of sequences is also refered here. Then we argue the condition when real normed subspaces become Banach’s spaces. We also formalize quotient vector space. In the last session, we argue the properties of the closure of real normed space. These formalizations are based on [19](p.3-41), [2] and [34](p.3-67).
7
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The Basic Existence Theorem of Riemann-Stieltjes Integral

81%
Nakasho K., Narita K., Shidama Y.
Formalized Mathematics
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2016
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tom 24
|
nr 4
253-259
EN
In this article, the basic existence theorem of Riemann-Stieltjes integral is formalized. This theorem states that if f is a continuous function and ρ is a function of bounded variation in a closed interval of real line, f is Riemann-Stieltjes integrable with respect to ρ. In the first section, basic properties of real finite sequences are formalized as preliminaries. In the second section, we formalized the existence theorem of the Riemann-Stieltjes integral. These formalizations are based on [15], [12], [10], and [11].
8
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Torsion Z-module and Torsion-free Z-module

81%
Futa Y., Okazaki H., Nakasho K., Shidama Y.
Formalized Mathematics
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2014
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tom 22
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nr 4
277-289
EN
In this article, we formalize a torsion Z-module and a torsionfree Z-module. Especially, we prove formally that finitely generated torsion-free Z-modules are finite rank free. We also formalize properties related to rank of finite rank free Z-modules. The notion of Z-module is necessary for solving lattice problems, LLL (Lenstra, Lenstra, and Lov´asz) base reduction algorithm [20], cryptographic systems with lattice [21], and coding theory [11].
9
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Riemann-Stieltjes Integral

81%
Narita K., Nakasho K., Shidama Y.
Formalized Mathematics
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2016
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tom 24
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nr 3
199-204
EN
In this article, the definitions and basic properties of Riemann-Stieltjes integral are formalized in Mizar [1]. In the first section, we showed the preliminary definition. We proved also some properties of finite sequences of real numbers. In Sec. 2, we defined variation. Using the definition, we also defined bounded variation and total variation, and proved theorems about related properties. In Sec. 3, we defined Riemann-Stieltjes integral. Referring to the way of the article [7], we described the definitions. In the last section, we proved theorems about linearity of Riemann-Stieltjes integral. Because there are two types of linearity in Riemann-Stieltjes integral, we proved linearity in two ways. We showed the proof of theorems based on the description of the article [7]. These formalizations are based on [8], [5], [3], and [4].
10
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Equivalent Expressions of Direct Sum Decomposition of Groups1

81%
Nakasho K., Okazaki H., Yamazaki H., Shidama Y.
Formalized Mathematics
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2015
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tom 23
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nr 1
67-73
EN
In this article, the equivalent expressions of the direct sum decomposition of groups are mainly discussed. In the first section, we formalize the fact that the internal direct sum decomposition can be defined as normal subgroups and some of their properties. In the second section, we formalize an equivalent form of internal direct sum of commutative groups. In the last section, we formalize that the external direct sum leads an internal direct sum. We referred to [19], [18] [8] and [14] in the formalization.
11
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σ-ring and σ-algebra of Sets1

81%
Endou N., Nakasho K., Shidama Y.
Formalized Mathematics
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2015
|
tom 23
|
nr 1
51-57
EN
In this article, semiring and semialgebra of sets are formalized so as to construct a measure of a given set in the next step. Although a semiring of sets has already been formalized in [13], that is, strictly speaking, a definition of a quasi semiring of sets suggested in the last few decades [15]. We adopt a classical definition of a semiring of sets here to avoid such a confusion. Ring of sets and algebra of sets have been formalized as non empty preboolean set [23] and field of subsets [18], respectively. In the second section, definitions of a ring and a σ-ring of sets, which are based on a semiring and a ring of sets respectively, are formalized and their related theorems are proved. In the third section, definitions of an algebra and a σ-algebra of sets, which are based on a semialgebra and an algebra of sets respectively, are formalized and their related theorems are proved. In the last section, mutual relationships between σ-ring and σ-algebra of sets are formalized and some related examples are given. The formalization is based on [15], and also referred to [9] and [16].
12
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Definition and Properties of Direct Sum Decomposition of Groups1

81%
Nakasho K., Yamazaki H., Okazaki H., Shidama Y.
Formalized Mathematics
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2015
|
tom 23
|
nr 1
15-27
EN
In this article, direct sum decomposition of group is mainly discussed. In the second section, support of element of direct product group is defined and its properties are formalized. It is formalized here that an element of direct product group belongs to its direct sum if and only if support of the element is finite. In the third section, product map and sum map are prepared. In the fourth section, internal and external direct sum are defined. In the last section, an equivalent form of internal direct sum is proved. We referred to [23], [22], [8] and [18] in the formalization.
13
Content available remote

Isomorphisms of Direct Products of Cyclic Groups of Prime Power Order

81%
Yamazaki H., Okazaki H., Nakasho K., Shidama Y.
Formalized Mathematics
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2013
|
tom 21
|
nr 3
207-211
EN
In this paper we formalized some theorems concerning the cyclic groups of prime power order. We formalize that every commutative cyclic group of prime power order is isomorphic to a direct product of family of cyclic groups [1], [18].
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