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# How Lipschitz Functions Characterize the Underlying Metric Spaces

• Lei Li,
School of Mathematical Sciences and LPMC, Nankai University, Tianjin, 300071, China
• Ya-Shu Wang,
Department of Mathematics, University of Alberta, Edmonton, AB, T6G 2G1
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## Abstract

Let $X, Y$ be metric spaces and $E, F$ be Banach spaces. Suppose that both $X,Y$ are realcompact, or both $E,F$ are realcompact. The zero set of a vector-valued function $f$ is denoted by $z(f)$. A linear bijection $T$ between local or generalized Lipschitz vector-valued function spaces is said to preserve zero-set containments or nonvanishing functions if $z(f)\subseteq z(g)\quad\Longleftrightarrow\quad z(Tf)\subseteq z(Tg),$ or $z(f) = \emptyset\quad \Longleftrightarrow\quad z(Tf)=\emptyset,$ respectively. Every zero-set containment preserver, and every nonvanishing function preserver when $\dim E =\dim F\lt +\infty$, is a weighted composition operator $(Tf)(y)=J_y(f(\tau(y)))$. We show that the map $\tau\colon Y\to X$ is a locally (little) Lipschitz homeomorphism.
 Keywords: (generalized, locally, little) Lipschitz functions, zero-set containment preservers, biseparating maps
 MSC Classifications: 46E40 - Spaces of vector- and operator-valued functions 54D60 - Realcompactness and realcompactification 46E15 - Banach spaces of continuous, differentiable or analytic functions