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How Lipschitz Functions Characterize the Underlying Metric Spaces

Published online by Cambridge University Press:  20 November 2018

Lei Li
Affiliation:
School of Mathematical Sciences and LPMC, Nankai University, Tianjin, 300071, China e-mail: leilee@nankai.edu.cn
Ya-Shu Wang
Affiliation:
Department of Mathematics, University of Alberta, Edmonton, AB, T6G 2G1 e-mail: yashu@ualberta.ca
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Abstract

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Let $X$ and $Y$ be metric spaces and $E$, $F$ be Banach spaces. Suppose that both $X$ and $Y$ are realcompact, or both $E$, $F$ are realcompact. The zero set of a vector-valued function $f$ is denoted by $z\left( f \right)$. A linear bijection $T$ between local or generalized Lipschitz vector-valued function spaces is said to preserve zero-set containments or nonvanishing functions if

$$z\left( f \right)\,\subseteq \,z\left( g \right)\,\,\,\,\Leftrightarrow \,\,\,\,z\left( Tf \right)\,\subseteq \,z\left( Tg \right),\,\,\,\,\,\text{or}\,\,\,\,z\left( f \right)\,=\,\varnothing \,\,\,\Leftrightarrow \,\,\,z\left( Tf \right)\,=\,\varnothing ,$$

respectively. Every zero-set containment preserver, and every nonvanishing function preserver when $\dim\,E\,=\,\dim\,F\,<\,+\infty$, is a weighted composition operator $\left( Tf \right)\left( y \right)\,=\,{{J}_{y}}\left( f\left( \tau \left( y \right) \right) \right)$. We show that the map $\tau \,:\,Y\,\to \,X$ is a locally (little) Lipschitz homeomorphism.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 2014

Footnotes

Li is supported by The National Natural Science Foundation of China (11271199)

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