How Lipschitz Functions Characterize the Underlying Metric Spaces

http://dx.doi.org/10.4153/CMB-2013-007-8
11 pages

 

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.