# Topological Properties of Real Normed Space

Formalized Mathematics (2014)

• Volume: 22, Issue: 3, page 209-223
• ISSN: 1426-2630

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## Abstract

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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).

## How to cite

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Kazuhisa Nakasho, Yuichi Futa, and Yasunari Shidama. "Topological Properties of Real Normed Space." Formalized Mathematics 22.3 (2014): 209-223. <http://eudml.org/doc/270985>.

@article{KazuhisaNakasho2014,
abstract = {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).},
author = {Kazuhisa Nakasho, Yuichi Futa, Yasunari Shidama},
journal = {Formalized Mathematics},
keywords = {functional analysis; normed linear space; topological vector space},
language = {eng},
number = {3},
pages = {209-223},
title = {Topological Properties of Real Normed Space},
url = {http://eudml.org/doc/270985},
volume = {22},
year = {2014},
}

TY - JOUR
AU - Kazuhisa Nakasho
AU - Yuichi Futa
AU - Yasunari Shidama
TI - Topological Properties of Real Normed Space
JO - Formalized Mathematics
PY - 2014
VL - 22
IS - 3
SP - 209
EP - 223
AB - 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).
LA - eng
KW - functional analysis; normed linear space; topological vector space
UR - http://eudml.org/doc/270985
ER -

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