Topological Properties of Real Normed Space

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).Nakasho Kazuhisa - Shinshu University Nagano, JapanFuta Yuichi - Japan Advanced Institute of Science and Technology Ishikawa, JapanShidama Yasunari - Shinshu University Nagano, JapanGrzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.Nicolas Bourbaki, H.G. Eggleston, and S. Madan. Elements of mathematics: Topological vector spaces. Springer-Verlag, 1987.Czesław Bylinski. The complex numbers. Formalized Mathematics, 1(3):507-513, 1990.Czesław Bylinski. Functions and their basic properties. Formalized Mathematics, 1(1): 55-65, 1990.Czesław Bylinski. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.Czesław Bylinski. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.Czesław Bylinski. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.Noboru Endou, Yasumasa Suzuki, and Yasunari Shidama. Real linear space of real sequences. Formalized Mathematics, 11(3):249-253, 2003.Noboru Endou, Yasunari Shidama, and Katsumasa Okamura. Baire’s category theorem and some spaces generated from real normed space. Formalized Mathematics, 14(4): 213-219, 2006. doi:10.2478/v10037-006-0024-x.Adam Grabowski. On the boundary and derivative of a set. Formalized Mathematics, 13 (1):139-146, 2005.Jarosław Kotowicz. Convergent real sequences. Upper and lower bound of sets of real numbers. Formalized Mathematics, 1(3):477-481, 1990.Jarosław Kotowicz. Quotient vector spaces and functionals. Formalized Mathematics, 11 (1):59-68, 2003.Eugeniusz Kusak, Wojciech Leonczuk, and Michał Muzalewski. Abelian groups, fields and vector spaces. Formalized Mathematics, 1(2):335-342, 1990.Keiko Narita, Noboru Endou, and Yasunari Shidama. Dual spaces and Hahn-Banach theorem. Formalized Mathematics, 22(1):69-77, 2014. doi:10.2478/forma-2014-0007.Takaya Nishiyama, Keiji Ohkubo, and Yasunari Shidama. The continuous functions on normed linear spaces. Formalized Mathematics, 12(3):269-275, 2004.Beata Padlewska. Families of sets. Formalized Mathematics, 1(1):147-152, 1990.Beata Padlewska and Agata Darmochwał. Topological spaces and continuous functions. Formalized Mathematics, 1(1):223-230, 1990.Jan Popiołek. Real normed space. Formalized Mathematics, 2(1):111-115, 1991.Walter Rudin. Functional Analysis. New York, McGraw-Hill, 2nd edition, 1991.Yasunari Shidama. Banach space of bounded linear operators. Formalized Mathematics, 12(1):39-48, 2004.Yasunari Shidama. The series on Banach algebra. Formalized Mathematics, 12(2):131-138, 2004.Yasumasa Suzuki, Noboru Endou, and Yasunari Shidama. Banach space of absolute summable real sequences. Formalized Mathematics, 11(4):377-380, 2003.Andrzej Trybulec. Domains and their Cartesian products. Formalized Mathematics, 1(1): 115-122, 1990.Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics, 1 (2):329-334, 1990.Andrzej Trybulec. On the sets inhabited by numbers. Formalized Mathematics, 11(4): 341-347, 2003.Wojciech A. Trybulec. Subspaces and cosets of subspaces in real linear space. Formalized Mathematics, 1(2):297-301, 1990.Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291-296, 1990.Wojciech A. Trybulec. Basis of real linear space. Formalized Mathematics, 1(5):847-850, 1990.Wojciech A. Trybulec. Subspaces and cosets of subspaces in vector space. Formalized Mathematics, 1(5):865-870, 1990.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73-83, 1990.Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990.Mirosław Wysocki and Agata Darmochwał. Subsets of topological spaces. Formalized Mathematics, 1(1):231-237, 1990.Kosaku Yoshida. Functional Analysis. Springer, 1980

Separability of Real Normed Spaces and Its Basic Properties

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].Kazuhisa Nakasho - Shinshu University, Nagano, JapanNoboru Endou - Gifu National College of Technology, Gifu, JapanJonathan Backer, Piotr Rudnicki, and Christoph Schwarzweller. Ring ideals. Formalized Mathematics, 9(3):565–582, 2001.Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377–382, 1990.Grzegorz Bancerek. Cardinal arithmetics. Formalized Mathematics, 1(3):543–547, 1990.Grzegorz Bancerek. König’s theorem. Formalized Mathematics, 1(3):589–593, 1990.Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41–46, 1990.Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91–96, 1990.Nicolas Bourbaki. Topological vector spaces: Chapters 1-5. Springer, 1981.Czesław Byliński. The complex numbers. Formalized Mathematics, 1(3):507–513, 1990.Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1): 55–65, 1990.Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153–164, 1990.Czesław Byliński. Partial functions. Formalized Mathematics, 1(2):357–367, 1990.Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47–53, 1990.Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165–167, 1990.N. J. Dunford and T. Schwartz. Linear operators I. Interscience Publ., 1958.Noboru Endou, Yasunari Shidama, and Katsumasa Okamura. Baire’s category theorem and some spaces generated from real normed space. Formalized Mathematics, 14(4): 213–219, 2006. doi:10.2478/v10037-006-0024-x.Andrey Kolmogorov and Sergei Fomin. Elements of the Theory of Functions and Functional Analysis [Two Volumes in One]. Martino Fine Books, 2012.Andrzej Kondracki. Basic properties of rational numbers. Formalized Mathematics, 1(5): 841–845, 1990.Eugeniusz Kusak, Wojciech Leończuk, and Michał Muzalewski. Abelian groups, fields and vector spaces. Formalized Mathematics, 1(2):335–342, 1990.Kazuhisa Nakasho, Yuichi Futa, and Yasunari Shidama. Topological properties of real normed space. Formalized Mathematics, 22(3):209–223, 2014. doi:10.2478/forma-2014-0024.Keiko Narita, Noboru Endou, and Yasunari Shidama. Dual spaces and Hahn-Banach theorem. Formalized Mathematics, 22(1):69–77, 2014. doi:10.2478/forma-2014-0007.Keiko Narita, Noboru Endou, and Yasunari Shidama. Bidual spaces and reflexivity of real normed spaces. Formalized Mathematics, 22(4):303–311, 2014. doi:10.2478/forma-2014-0030.Bogdan Nowak and Andrzej Trybulec. Hahn-Banach theorem. Formalized Mathematics, 4(1):29–34, 1993.Beata Padlewska and Agata Darmochwał. Topological spaces and continuous functions. Formalized Mathematics, 1(1):223–230, 1990.Jan Popiołek. Real normed space. Formalized Mathematics, 2(1):111–115, 1991.Yasunari Shidama. Banach space of bounded linear operators. Formalized Mathematics, 12(1):39–48, 2004.Andrzej Trybulec. Domains and their Cartesian products. Formalized Mathematics, 1(1): 115–122, 1990.Wojciech A. Trybulec. Subspaces and cosets of subspaces in real linear space. Formalized Mathematics, 1(2):297–301, 1990.Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291–296, 1990.Wojciech A. Trybulec. Linear combinations in real linear space. Formalized Mathematics, 1(3):581–588, 1990.Wojciech A. Trybulec. Basis of real linear space. Formalized Mathematics, 1(5):847–850, 1990.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67–71, 1990.Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73–83, 1990.Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181–186, 1990.Kosaku Yoshida. Functional Analysis. Springer, 1980

Bidual Spaces and Reflexivity of Real Normed Spaces

In this article, we considered bidual spaces and reflexivity of real normed spaces. At first we proved some corollaries applying Hahn-Banach theorem and showed related theorems. In the second section, we proved the norm of dual spaces and defined the natural mapping, from real normed spaces to bidual spaces. We also proved some properties of this mapping. Next, we defined real normed space of R, real number spaces as real normed spaces and proved related theorems. We can regard linear functionals as linear operators by this definition. Accordingly we proved Uniform Boundedness Theorem for linear functionals using the theorem (5) from [21]. Finally, we defined reflexivity of real normed spaces and proved some theorems about isomorphism of linear operators. Using them, we proved some properties about reflexivity. These formalizations are based on [19], [20], [8] and [1].Narita Keiko - Hirosaki-city Aomori, JapanEndou Noboru - Gifu National College of Technology Gifu, JapanShidama Yasunari - Shinshu University Nagano, JapanHaim Brezis. Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer, 2011.Czesław Bylinski. The complex numbers. Formalized Mathematics, 1(3):507-513, 1990.Czesław Bylinski. Functions and their basic properties. Formalized Mathematics, 1(1): 55-65, 1990.Czesław Bylinski. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.Czesław Bylinski. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.Czesław Bylinski. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.Agata Darmochwał. The Euclidean space. Formalized Mathematics, 2(4):599-603, 1991.Peter D. Dax. Functional Analysis. Pure and Applied Mathematics: A Wiley Series of Texts, Monographs and Tracts. Wiley Interscience, 2002.Noboru Endou, Yasunari Shidama, and Katsumasa Okamura. Baire’s category theorem and some spaces generated from real normed space. Formalized Mathematics, 14(4): 213-219, 2006. doi:10.2478/v10037-006-0024-x.Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1): 35-40, 1990.Jarosław Kotowicz. Convergent real sequences. Upper and lower bound of sets of real numbers. Formalized Mathematics, 1(3):477-481, 1990.Eugeniusz Kusak, Wojciech Leonczuk, and Michał Muzalewski. Abelian groups, fields and vector spaces. Formalized Mathematics, 1(2):335-342, 1990.Kazuhisa Nakasho, Yuichi Futa, and Yasunari Shidama. Topological properties of real normed space. Formalized Mathematics, 22(3):209-223, 2014. doi:10.2478/forma-2014-0024.Keiko Narita, Noboru Endou, and Yasunari Shidama. Dual spaces and Hahn-Banach theorem. Formalized Mathematics, 22(1):69-77, 2014. doi:10.2478/forma-2014-0007.Takaya Nishiyama, Keiji Ohkubo, and Yasunari Shidama. The continuous functions on normed linear spaces. Formalized Mathematics, 12(3):269-275, 2004.Bogdan Nowak and Andrzej Trybulec. Hahn-Banach theorem. Formalized Mathematics, 4(1):29-34, 1993.Jan Popiołek. Some properties of functions modul and signum. Formalized Mathematics, 1(2):263-264, 1990.Jan Popiołek. Real normed space. Formalized Mathematics, 2(1):111-115, 1991.Michael Reed and Barry Simon. Methods of modern mathematical physics. Vol. 1. Academic Press, New York, 1972.Walter Rudin. Functional Analysis. New York, McGraw-Hill, 2nd edition, 1991.Hideki Sakurai, Hisayoshi Kunimune, and Yasunari Shidama. Uniform boundedness principle. Formalized Mathematics, 16(1):19-21, 2008. doi:10.2478/v10037-008-0003-5.Yasunari Shidama. Banach space of bounded linear operators. Formalized Mathematics, 12(1):39-48, 2004.Yasumasa Suzuki, Noboru Endou, and Yasunari Shidama. Banach space of absolute summable real sequences. Formalized Mathematics, 11(4):377-380, 2003.Andrzej Trybulec. On the sets inhabited by numbers. Formalized Mathematics, 11(4): 341-347, 2003.Wojciech A. Trybulec. Subspaces and cosets of subspaces in real linear space. Formalized Mathematics, 1(2):297-301, 1990.Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291-296, 1990.Wojciech A. Trybulec. Subspaces of real linear space generated by one, two, or three vectors and their cosets. Formalized Mathematics, 3(2):271-274, 1992.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73-83, 1990

Weak Convergence and Weak Convergence

AbstractIn this article, we deal with weak convergence on sequences in real normed spaces, and weak* convergence on sequences in dual spaces of real normed spaces. In the first section, we proved some topological properties of dual spaces of real normed spaces. We used these theorems for proofs of Section 3. In Section 2, we defined weak convergence and weak* convergence, and proved some properties. By RNS_Real Mizar functor, real normed spaces as real number spaces already defined in the article [18], we regarded sequences of real numbers as sequences of RNS_Real. So we proved the last theorem in this section using the theorem (8) from [25]. In Section 3, we defined weak sequential compactness of real normed spaces. We showed some lemmas for the proof and proved the theorem of weak sequential compactness of reflexive real Banach spaces. We referred to [36], [23], [24] and [3] in the formalization.This work was supported by JSPS KAKENHI 22300285 and 2350002.Keiko Narita - Hirosaki-city, Aomori, JapanYasunari Shidama - Shinshu University, Nagano, JapanNoboru Endou - Gifu National College of Technology, Gifu, JapanGrzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.Haim Brezis. Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer, 2011.Czesław Bylinski. The complex numbers. Formalized Mathematics, 1(3):507-513, 1990.Czesław Bylinski. Functions and their basic properties. Formalized Mathematics, 1(1): 55-65, 1990.Czesław Bylinski. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.Czesław Bylinski. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.Czesław Bylinski. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.Noboru Endou, Yasunari Shidama, and Katsumasa Okamura. Baire’s category theorem and some spaces generated from real normed space. Formalized Mathematics, 14(4): 213-219, 2006. doi:10.2478/v10037-006-0024-x. [Crossref]Krzysztof Hryniewiecki. Recursive definitions. Formalized Mathematics, 1(2):321-328, 1990.Artur Korniłowicz. Recursive definitions. Part II. Formalized Mathematics, 12(2):167-172, 2004.Jarosław Kotowicz. Monotone real sequences. Subsequences. Formalized Mathematics, 1 (3):471-475, 1990.Jarosław Kotowicz. Real sequences and basic operations on them. Formalized Mathematics, 1(2):269-272, 1990.Jarosław Kotowicz. Convergent sequences and the limit of sequences. Formalized Mathematics, 1(2):273-275, 1990.Jarosław Kotowicz. Convergent real sequences. Upper and lower bound of sets of real numbers. Formalized Mathematics, 1(3):477-481, 1990.Kazuhisa Nakasho, Yuichi Futa, and Yasunari Shidama. Topological properties of real normed space. Formalized Mathematics, 22(3):209-223, 2014. doi:10.2478/forma-2014-0024. [Crossref]Keiko Narita, Noboru Endou, and Yasunari Shidama. Dual spaces and Hahn-Banach theorem. Formalized Mathematics, 22(1):69-77, 2014. doi:10.2478/forma-2014-0007. [Crossref]Keiko Narita, Noboru Endou, and Yasunari Shidama. Bidual spaces and reflexivity of real normed spaces. Formalized Mathematics, 22(4):303-311, 2014. doi:10.2478/forma-2014-0030. [Crossref]Adam Naumowicz. Conjugate sequences, bounded complex sequences and convergent complex sequences. Formalized Mathematics, 6(2):265-268, 1997.Takaya Nishiyama, Keiji Ohkubo, and Yasunari Shidama. The continuous functions on normed linear spaces. Formalized Mathematics, 12(3):269-275, 2004.Bogdan Nowak and Andrzej Trybulec. Hahn-Banach theorem. Formalized Mathematics, 4(1):29-34, 1993.Jan Popiołek. Real normed space. Formalized Mathematics, 2(1):111-115, 1991.Michael Reed and Barry Simon. Methods of modern mathematical physics. Vol. 1. Academic Press, New York, 1972.Walter Rudin. Functional Analysis. New York, McGraw-Hill, 2nd edition, 1991.Hideki Sakurai, Hisayoshi Kunimune, and Yasunari Shidama. Uniform boundedness principle. Formalized Mathematics, 16(1):19-21, 2008. doi:10.2478/v10037-008-0003-5. [Crossref]Yasunari Shidama. Banach space of bounded linear operators. Formalized Mathematics, 12(1):39-48, 2004.Yasumasa Suzuki, Noboru Endou, and Yasunari Shidama. Banach space of absolute summable real sequences. Formalized Mathematics, 11(4):377-380, 2003.Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics, 1 (2):329-334, 1990.Andrzej Trybulec. On the sets inhabited by numbers. Formalized Mathematics, 11(4): 341-347, 2003.Michał J. Trybulec. Integers. Formalized Mathematics, 1(3):501-505, 1990.Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291-296, 1990.Wojciech A. Trybulec. Basis of real linear space. Formalized Mathematics, 1(5):847-850, 1990.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73-83, 1990.Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990.Kosaku Yoshida. Functional Analysis. Springer, 1980.Bo Zhang, Hiroshi Yamazaki, and Yatsuka Nakamura. Inferior limit and superior limit of sequences of real numbers. Formalized Mathematics, 13(3):375-381, 2005. [Web of Science