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How Lipschitz Functions Characterize the Underlying Metric Spaces


Published:20130526
Printed: Jun 2014
Lei Li,
School of Mathematical Sciences and LPMC, Nankai University, Tianjin, 300071, China
YaShu 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 vectorvalued function $f$ is denoted by $z(f)$.
A linear bijection $T$ between local or generalized Lipschitz vectorvalued function spaces
is said to preserve zeroset 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 zeroset 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.