Object's details: Compactness in normed spaces: a unified approach through semi-norms

Provider:Akademicka Platforma Czasopism

Description

Title:
Compactness in normed spaces: a unified approach through semi-norms
Creator:
Gulgowski, Jacek;
Kasprzak, Piotr;
Maćkowiak, Piotr
Description:
In this paper we prove two new abstract compactness criteria in normed spaces. To this end we first introduce the notion of an equinormed set using a suitable family of semi-norms on the given normed space satisfying some natural conditions. Those conditions, roughly speaking, state that the norm can be approximated (on the equinormed sets even uniformly) by the elements of this family. As we are given some freedom of choice of the underlying semi-normed structure that is used to define equinormed sets, our approach opens a new perspective for building compactness criteria in specific normed spaces. As an example we show that natural selections of families of semi-norms in spaces $C(X,\R)$ and $l^p$ for $p\in[1,+\infty)$ lead to the well-known compactness criteria (including the Arzel\`a-Ascoli theorem). In the second part of the paper, applying the abstract theorems, we construct a simple compactness criterion in the space of functions of bounded Schramm variation.
Object availability:
free access
Rights: https://creativecommons.org/licenses/by-nd/4.0 ;
Prawa autorskie (c) 2023 Jacek Gulgowski, Piotr Kasprzak, Piotr Maćkowiak
Date:
2023.09.23
Type:
info:eu-repo/semantics/article;
info:eu-repo/semantics/publishedVersion;
Recenzowany artykuł
Language:
English
Source:
Topological Methods in Nonlinear Analysis; Vol 62, No 1 (September 2023); 105 - 134;
1230-3429
Publisher:
Nicolaus Copernicus University in Toruń
Relation: https://apcz.umk.pl/TMNA/article/view/46309/36555 ;https://apcz.umk.pl/TMNA/article/view/46309/36556
Subject:
Compactness criterion;
equinormed set;
functions of bounded Schramm variation;
precompact set;
relatively compact set;
semi-norm;
Primary: 46B50;
Secondary: 26A45
Identifier: https://apcz.umk.pl/TMNA/article/view/46309
Data provider:
Akademicka Platforma Czasopism

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