Generalized isometry in normed spaces
Commun. Korean Math. Soc.
Published online June 10, 2021
Abbas Zivari-Kazempour
Department of Mathematics, University of Ayatollah Borujerdi, Borujerd, Iran
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 : 46H40; 47A10
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