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Formalized Mathematics

(a computer assisted approach)

Editor-in-Chief: Matuszewski, Roman

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Volume 23, Issue 1 (Mar 2015)

Issues

Separability of Real Normed Spaces and Its Basic Properties

Kazuhisa Nakasho
  • Shinshu University, Nagano, Japan
/ Noboru Endou
  • Gifu National College of Technology, Gifu, Japan
Published Online: 2015-03-31 | DOI: https://doi.org/10.2478/forma-2015-0005

Summary

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

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About the article

Received: 2015-02-26

Published Online: 2015-03-31

Published in Print: 2015-03-01



Citation Information: Formalized Mathematics, 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. (CC BY-SA 3.0)

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