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# Isometry on Linear $n$-G-quasi Normed Spaces

Published:2016-11-03
Printed: Jun 2017
• Yumei Ma,
Department of Mathematics, Dalian Nationalities University, 116600 Dalian, China
 Format: LaTeX MathJax PDF

## Abstract

This paper generalizes the Aleksandrov problem: the Mazur-Ulam theorem on $n$-G-quasi normed spaces. It proves that a one-$n$-distance preserving mapping is an $n$-isometry if and only if it has the zero-$n$-G-quasi preserving property, and two kinds of $n$-isometries on $n$-G-quasi normed space are equivalent; we generalize the Benz theorem to n-normed spaces with no restrictions on the dimension of spaces.
 Keywords: $n$-G-quasi norm, Mazur-Ulam theorem, Aleksandrov problem, $n$-isometry, $n$-0-distance
 MSC Classifications: 46B20 - Geometry and structure of normed linear spaces 46B04 - Isometric theory of Banach spaces 51K05 - General theory

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