Generalized isometry in normed spaces
Commun. Korean Math. Soc. 2022 Vol. 37, No. 1, 105-112
https://doi.org/10.4134/CKMS.c200451
Published online June 10, 2021
Printed January 31, 2022
Abbas Zivari-Kazempour
Ayatollah Borujerdi University
Abstract : Let $g:X\longrightarrow Y$ and $f:Y\longrightarrow Z$ 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
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