Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Formalized Mathematics

(a computer assisted approach)

Editor-in-Chief: Matuszewski, Roman

4 Issues per year

SCImago Journal Rank (SJR) 2016: 0.207
Source Normalized Impact per Paper (SNIP) 2016: 0.315

Open Access
See all formats and pricing
More options …
Volume 23, Issue 1


Separability of Real Normed Spaces and Its Basic Properties

Kazuhisa Nakasho / Noboru Endou
Published Online: 2015-03-31 | DOI: https://doi.org/10.2478/forma-2015-0005


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].

MSC: 46B20; 46A19; 03B35

Keywords: functional analysis; normed linear space; topological vector space

MML: identifier: NORMSP _4; version: 8.1.04 5.31.1231


  • [1] Jonathan Backer, Piotr Rudnicki, and Christoph Schwarzweller. Ring ideals. Formalized Mathematics, 9(3):565–582, 2001.Google Scholar

  • [2] Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377–382, 1990.Google Scholar

  • [3] Grzegorz Bancerek. Cardinal arithmetics. Formalized Mathematics, 1(3):543–547, 1990.Google Scholar

  • [4] Grzegorz Bancerek. König’s theorem. Formalized Mathematics, 1(3):589–593, 1990.Google Scholar

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

  • [6] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91–96, 1990.Google Scholar

  • [7] Nicolas Bourbaki. Topological vector spaces: Chapters 1-5. Springer, 1981.Google Scholar

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

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

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

  • [11] Czesław Byliński. Partial functions. Formalized Mathematics, 1(2):357–367, 1990.Google Scholar

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

  • [13] Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165–167, 1990.Google Scholar

  • [14] N. J. Dunford and T. Schwartz. Linear operators I. Interscience Publ., 1958.Google Scholar

  • [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.CrossrefGoogle Scholar

  • [16] Andrey Kolmogorov and Sergei Fomin. Elements of the Theory of Functions and Functional Analysis [Two Volumes in One]. Martino Fine Books, 2012.Google Scholar

  • [17] Andrzej Kondracki. Basic properties of rational numbers. Formalized Mathematics, 1(5): 841–845, 1990.Google Scholar

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

  • [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.CrossrefGoogle Scholar

  • [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.CrossrefGoogle Scholar

  • [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.CrossrefGoogle Scholar

  • [22] Bogdan Nowak and Andrzej Trybulec. Hahn-Banach theorem. Formalized Mathematics, 4(1):29–34, 1993.Google Scholar

  • [23] Beata Padlewska and Agata Darmochwał. Topological spaces and continuous functions. Formalized Mathematics, 1(1):223–230, 1990.Google Scholar

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

  • [25] Yasunari Shidama. Banach space of bounded linear operators. Formalized Mathematics, 12(1):39–48, 2004.Google Scholar

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

  • [27] Wojciech A. Trybulec. Subspaces and cosets of subspaces in real linear space. Formalized Mathematics, 1(2):297–301, 1990.Google Scholar

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

  • [29] Wojciech A. Trybulec. Linear combinations in real linear space. Formalized Mathematics, 1(3):581–588, 1990.Google Scholar

  • [30] Wojciech A. Trybulec. Basis of real linear space. Formalized Mathematics, 1(5):847–850, 1990.Google Scholar

  • [31] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67–71, 1990.Google Scholar

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

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

  • [34] Kosaku Yoshida. Functional Analysis. Springer, 1980.Google Scholar

About the article

Received: 2015-02-26

Published Online: 2015-03-31

Published in Print: 2015-03-01

Citation Information: Formalized Mathematics, Volume 23, Issue 1, Pages 59–65, ISSN (Online) 1898-9934, DOI: https://doi.org/10.2478/forma-2015-0005.

Export Citation

© Kazuhisa Nakasho et al.. This work is licensed under the Creative Commons Attribution-ShareAlike 3.0 License. BY-SA 3.0

Comments (0)

Please log in or register to comment.
Log in