Isometries, Mazur-Ulam theorem and Aleksandrov problem for non-Archimedean normed spaces
type: Item
instance of: scholarly article
title: Isometries, Mazur-Ulam theorem and Aleksandrov problem for non-Archimedean normed spaces
zbMATH Open document ID: 1248.46050
author: A. Kubzdela
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2012-05-04T00:00:00Z
reviewed by: Cristina Pérez-García
Mathematics Subject Classification ID: 46S10
Mathematics Subject Classification ID: 47S10
zbMATH DE Number: 6030594
zbMATH Keywords: isometry
zbMATH Keywords: Mazur-Ulam theorem
zbMATH Keywords: non-Archimedean normed space
zbMATH Keywords: conservative distance
MaRDI profile type: Publication
full work available at URL: j.na.2011.10.006
OpenAlex ID: W2065626815
cites work: Q4333558
cites work: Properties of isometric mappings
cites work: Isometries in Normed Spaces
cites work: A Mazur--Ulam theorem in non-Archimedean normed spaces
cites work: Q5615895
cites work: On Isometries of Euclidean Spaces
cites work: On the Aleksandrov Problem of Conservative Distances
cites work: On the Mazur-Ulam Theorem and the Aleksandrov Problem for Unit Distance Preserving Mappings
cites work: Q3184845
cites work: On the Aleksandrov-Rassias problem and the Hyers-Ulam-Rassias stability problem
cites work: Q3322680
cites work: Q4179411
cites work: Q3401901
DOI: 10.1016/J.NA.2011.10.006
review text: This paper deals with isometries in non-Archimedean normed spaces. Its main purpose is to study the behaviour in the non-Archimedean setting of the classical Mazur-Ulam theorem and the classical Aleksandrov problem, both extensively studied in the Archimedean literature.NEWLINENEWLINES. Mazur and S. Ulam proved in 1932 that if \(E,F\) are real normed spaces and \(T: E \rightarrow F\) is a surjective isometry then \(f\) is a linear map up to translation. Now, let us replace the field \(\mathbb{R}\) of real numbers by a complete non-Archimedean non-trivially valued field \(\mathbb{K}\). A non-Archimedean version of this theorem, for normed spaces over \(\mathbb{K}\), was given by \textit{M. S. Moslehian} and \textit{G. A. Sadeghi} [Nonlinear Anal., Theory Methods Appl. 69, No. 10, A, 3405--3408 (2008; Zbl 1160.46049)], showing also that the Marzur-Ulam theorem fails in the non-Archimedean case. In the present paper, the author proves the following stronger result: Let \(E,F\) be normed spaces over \(\mathbb{K}\). Assume that there exists a surjective isometry \(E \rightarrow F\). If every surjective isometry \(E \rightarrow F\) is an additive map up to translation, then \(E= F = \{ 0 \}\).NEWLINENEWLINEAlso, in 1970, A. D. Aleksandrov asked under what conditions a map of a metric space into itself satisfying the strong distance one preserving property (SDOPP) is an isometry (recall that if \(E,F\) are normed spaces, a map \(T: E \rightarrow F\) has the SDOPP if, for all \(x,y \in E\), one verifies \(\| x - y \| = 1\) if and only if \(\| T(x) - T(y )\| = 1\)). \textit{T. M. Rassias} and \textit{P. Šemrl} proved in [Proc. Am. Math. Soc. 118, No. 3, 919--925 (1993; Zbl 0780.51010)] that if \(T: E \rightarrow F\) is a map between real normed spaces \(E,F\) such that one of them has dimension \(>1\) andNEWLINENEWLINE(1) \(T\) is surjective,NEWLINENEWLINE(2) \(T\) is a \(1\)-Lipschitz map (i.e., \(\| T(x) - T(y) \| \leq \| x-y \|\) for all \(x,y \in E\)),NEWLINENEWLINE(3) \…
description: scientific article; zbMATH DE number 6030594