Connections between the completion of normed spaces over non-Archimedean fields and the stability of the Cauchy equation
type: Item
instance of: scholarly article
title: Connections between the completion of normed spaces over non-Archimedean fields and the stability of the Cauchy equation
zbMATH Open document ID: 1466.39020
author: Jens Schwaiger
published in: Annales Mathematicae Silesianae
publication date: 2021-05-03T00:00:00Z
review text: The concept of stability of a functional equation arises when one replaces a functional equation by an inequality which acts as a perturbation of the equation. The problem of stability of functional equations was motivated by a question of Ulam being asked in 1940. \textit{M. S. Moslehian} and \textit{T. M. Rassias} [Appl. Anal. Discrete Math. 1, No. 2, 325--334 (2007; Zbl 1257.39019)] proved stability of some functional equations in non-Archimedean spaces. Here the authors provide a close connection between stability results for the Cauchy equation and the completion of a normed space over the rationals endowed with the usual absolute value. In this paper similar results are presented when the valuation of the rationals is a \(p\)-adic valuation.
reviewed by: Ghadir Sadeghi
Mathematics Subject Classification ID: 39B82
Mathematics Subject Classification ID: 39B52
Mathematics Subject Classification ID: 46S10
zbMATH DE Number: 7342913
zbMATH Keywords: Cauchy equation
zbMATH Keywords: Hyers-Ulam stability
zbMATH Keywords: \(p\)-adic functional analysis
MaRDI profile type: Publication
full work available at URL: amsil-2020-0002
OpenAlex ID: W3029540217
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DOI: 10.2478/AMSIL-2020-0002
description: scientific article; zbMATH DE number 7342913