On the Cauchy difference on normed spaces
type: Item
instance of: scholarly article
title: On the Cauchy difference on normed spaces
zbMATH Open document ID: 0864.39009
DOI: 10.1007/BF02940800
published in: Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg
publication date: 1997-06-22T00:00:00Z
review text: Let \(f\) be a function from a real linear normed space \(X\) into a topological group \((G,+)\). Fixed a discrete normal subgroup \(K\) of \(G\), one may consider the condition \(f(x+y)- f(x)- f(y) \in K\) whenever \(|x|= |y|\). The author of this paper studies this condition and finds several theorems showing that under some regularity assumptions on \(f\) (continuity at a point, measurability in a precise sense, \(\dots)\) and some natural assumptions on \(G,K\) and \(X\), there exists a continuous additive function \(A\) from \(X\) into \(G\) such that \(f(x)-A(x)\in K\).
Mathematics Subject Classification ID: 39B52
zbMATH DE Number: 977862
zbMATH Keywords: conditional functional equation
zbMATH Keywords: Cauchy equation
zbMATH Keywords: stability
zbMATH Keywords: normed space
zbMATH Keywords: topological group
author: Janusz Brzdȩk
reviewed by: Claudi Alsina
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02940800
OpenAlex ID: W2084372244
description: scientific article; zbMATH DE number 977862