How Lipschitz Functions Characterize the Underlying Metric Spaces

Li, Lei ; Wang, Ya-Shu

Canadian mathematical bulletin, Tome 57 (2014) no. 2, pp. 364-374

Résumé

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.

Détail

DOI : 10.4153/CMB-2013-007-8

Mots-clés : 46E40, 54D60, 46E15, (generalized, locally, little) Lipschitz functions, zero-set containment preservers, biseparating maps