The singular case in the stability of additive functions
type: Item
instance of: scholarly article
title: The singular case in the stability of additive functions
zbMATH Open document ID: 1006.46002
author: Félix Cabello Sánchez
published in: Journal of Mathematical Analysis and Applications
publication date: 2002-07-04T00:00:00Z
review text: A mapping \(\omega: Z\mapsto Y\) acting between quasi-normed spaces is quasi-additive if it satisfies an estimate \(\|\omega (x+y)-\omega (x)-\omega (y)\|_Y\leq\varepsilon (\|x\|_Z +\|y\|_Z)\) for some \(\varepsilon >0\) and all \(x,y\in Z\). This mapping \(\omega\) is said to be approximable if there is an additive map \(a: Z\mapsto Y\) such that \(\omega-a\) is continuous at the origin of \(Z\). Assuming that \(Z\) is a finite dimensional space and \(Y\) is a quasi-Banach space, it is proved that every quasi-additive map \(\omega: Z\mapsto Y\) is approximable. In the case when both \(Y\) and \(Z\) are possibly infinite dimensional quasi-Banach spaces it is proved that every quasi-additive map \(\omega: Z\mapsto Y\) is approximable if and only if every homogeneous quasi-additive map \(\omega: Z\mapsto Y\) is asymptotically linear.
Mathematics Subject Classification ID: 46A16
Mathematics Subject Classification ID: 46A99
zbMATH DE Number: 1763472
zbMATH Keywords: asymptotically linear
zbMATH Keywords: quasi-Banach space
zbMATH Keywords: quasi-normed spaces
zbMATH Keywords: approximable
zbMATH Keywords: quasi-additive map
zbMATH Keywords: homogeneous
reviewed by: Rimas Norvaiša
MaRDI profile type: Publication
OpenAlex ID: W2021692336
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DOI: 10.1006/JMAA.2001.7759
description: scientific article; zbMATH DE number 1763472