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How Lipschitz Functions Characterize the Underlying Metric Spaces
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- Journal:
- Canadian Mathematical Bulletin / Volume 57 / Issue 2 / 14 June 2014
- Published online by Cambridge University Press:
- 20 November 2018, pp. 364-374
- Print publication:
- 14 June 2014
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- Article
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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.
