Quasi-additive mappings.
type: Item
instance of: scholarly article
title: Quasi-additive mappings.
zbMATH Open document ID: 1059.46003
author: Félix Cabello Sánchez
published in: Journal of Mathematical Analysis and Applications
publication date: 2004-03-14T00:00:00Z
review text: A map \(f\) between Abelian topological groups is called quasi-additive if the function of two variables \(f(x+y)-f(x)-f(y)\) is continuous at the origin. Obvious examples are additive maps, maps which are continuous at the origin, and sums of such maps. The author proves that there are no others, in case the range is a quasi-Banach space, and the domain is a vector space equipped with a Hausdorff weak topology. This unifies several known results, with a proof which is no more technical than those existing in the literature. An important special case is when the domain is the real line. The same conclusion also holds if the range is a quasi-Banach space, and the domain is the space of measurable functions on a finite measure space, equipped with the topology of convergence in measure. An application is that, within the category of Abelian topological groups, any extension of a quasi-Banach space by a quasi-Banach space is again a quasi-Banach space. It remains open whether these results can be generalized from quasi-Banach spaces to \(F\)-spaces.
Mathematics Subject Classification ID: 46A16
Mathematics Subject Classification ID: 39B82
Mathematics Subject Classification ID: 47H99
zbMATH DE Number: 2055377
zbMATH Keywords: quasi-Banach space
zbMATH Keywords: quasi-additivity
MaRDI profile type: Publication
full work available at URL: j.jmaa.2003.09.077
OpenAlex ID: W2069325631
cites work: The singular case in the stability of additive functions
cites work: Three-space problems in Banach space theory
cites work: On the splitting of twisted sums, and the three space problem for local convexity
cites work: On the Stability of the Linear Functional Equation
cites work: On the asymptoticity aspect of Hyers-Ulam stability of mappings
cites work: Q4178058
cites work: Q4175993
cites work: Q3219291
cites work: An Introduction to Banach Space Theory
cites work: On the three-space-problem for topological vector spaces
cites work: On Riemann integration of functions with values in topological linear spaces
cites work: Q4297314
DOI: 10.1016/J.JMAA.2003.09.077
description: scientific article; zbMATH DE number 2055377