Communications of the
Korean Mathematical Society
CKMS

ISSN(Print) 1225-1763 ISSN(Online) 2234-3024

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Commun. Korean Math. Soc. 2022; 37(1): 105-112

Online first article June 10, 2021      Printed January 31, 2022

https://doi.org/10.4134/CKMS.c200451

Copyright © The Korean Mathematical Society.

Generalized isometry in normed spaces

Abbas Zivari-Kazempour

Ayatollah Borujerdi University

Abstract

Let $g:Xlongrightarrow Y$ and $f:Ylongrightarrow Z$ be two maps between real normed linear spaces. Then $f$ is called generalized isometry or $g$-isometry if for each $x,y in X$, $$ Vert f(g(x))-f(g(y))Vert=Vert g(x)-g(y)Vert. $$ In this paper, under special hypotheses, we prove that each generalized isometry is affine. Some examples of generalized isometry are given as well.

Keywords: Isometry, Mazur-Ulam theorem, strictly convex, affine map

MSC numbers: Primary 46H40, 47A10