# Separability of Real Normed Spaces and Its Basic Properties

Formalized Mathematics (2015)

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

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## Abstract

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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].

## How to cite

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Kazuhisa 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 -

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