# Separability of Real Normed Spaces and Its Basic Properties

Kazuhisa Nakasho; Noboru Endou

Formalized Mathematics (2015)

- Volume: 23, Issue: 1, page 59-65
- ISSN: 1426-2630

## Access Full Article

top## Abstract

top## How to cite

topKazuhisa Nakasho, and Noboru Endou. "Separability of Real Normed Spaces and Its Basic Properties." Formalized Mathematics 23.1 (2015): 59-65. <http://eudml.org/doc/270951>.

@article{KazuhisaNakasho2015,

abstract = {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].},

author = {Kazuhisa Nakasho, Noboru Endou},

journal = {Formalized Mathematics},

keywords = {functional analysis; normed linear space; topological vector space},

language = {eng},

number = {1},

pages = {59-65},

title = {Separability of Real Normed Spaces and Its Basic Properties},

url = {http://eudml.org/doc/270951},

volume = {23},

year = {2015},

}

TY - JOUR

AU - Kazuhisa Nakasho

AU - Noboru Endou

TI - Separability of Real Normed Spaces and Its Basic Properties

JO - Formalized Mathematics

PY - 2015

VL - 23

IS - 1

SP - 59

EP - 65

AB - 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].

LA - eng

KW - functional analysis; normed linear space; topological vector space

UR - http://eudml.org/doc/270951

ER -

## References

top- [1] Jonathan Backer, Piotr Rudnicki, and Christoph Schwarzweller. Ring ideals. Formalized Mathematics, 9(3):565–582, 2001.
- [2] Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377–382, 1990.
- [3] Grzegorz Bancerek. Cardinal arithmetics. Formalized Mathematics, 1(3):543–547, 1990.
- [4] Grzegorz Bancerek. König’s theorem. Formalized Mathematics, 1(3):589–593, 1990.
- [5] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41–46, 1990. Zbl06213858
- [6] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91–96, 1990.
- [7] Nicolas Bourbaki. Topological vector spaces: Chapters 1-5. Springer, 1981. Zbl1106.46003
- [8] Czesław Byliński. The complex numbers. Formalized Mathematics, 1(3):507–513, 1990.
- [9] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1): 55–65, 1990.
- [10] Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153–164, 1990.
- [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] N. J. Dunford and T. Schwartz. Linear operators I. Interscience Publ., 1958. Zbl0084.10402
- [15] 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.
- [16] Andrey Kolmogorov and Sergei Fomin. Elements of the Theory of Functions and Functional Analysis [Two Volumes in One]. Martino Fine Books, 2012.
- [17] Andrzej Kondracki. Basic properties of rational numbers. Formalized Mathematics, 1(5): 841–845, 1990.
- [18] Eugeniusz Kusak, Wojciech Leończuk, and Michał Muzalewski. Abelian groups, fields and vector spaces. Formalized Mathematics, 1(2):335–342, 1990.
- [19] 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. Zbl1311.46016
- [20] 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. Zbl1298.46005
- [21] 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. Zbl1316.46014
- [22] Bogdan Nowak and Andrzej Trybulec. Hahn-Banach theorem. Formalized Mathematics, 4(1):29–34, 1993.
- [23] Beata Padlewska and Agata Darmochwał. Topological spaces and continuous functions. Formalized Mathematics, 1(1):223–230, 1990.
- [24] Jan Popiołek. Real normed space. Formalized Mathematics, 2(1):111–115, 1991.
- [25] Yasunari Shidama. Banach space of bounded linear operators. Formalized Mathematics, 12(1):39–48, 2004.
- [26] Andrzej Trybulec. Domains and their Cartesian products. Formalized Mathematics, 1(1): 115–122, 1990.
- [27] Wojciech A. Trybulec. Subspaces and cosets of subspaces in real linear space. Formalized Mathematics, 1(2):297–301, 1990.
- [28] Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291–296, 1990.
- [29] Wojciech A. Trybulec. Linear combinations in real linear space. Formalized Mathematics, 1(3):581–588, 1990.
- [30] Wojciech A. Trybulec. Basis of real linear space. Formalized Mathematics, 1(5):847–850, 1990.
- [31] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67–71, 1990.
- [32] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73–83, 1990.
- [33] Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181–186, 1990.
- [34] Kosaku Yoshida. Functional Analysis. Springer, 1980.

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.