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
\[z(f)\subseteq z(g)\quad\Longleftrightarrow\quad z(Tf)\subseteq z(Tg),\]
\[z(f) = \emptyset\quad \Longleftrightarrow\quad z(Tf)=\emptyset,\]
Every zero-set containment preserver, and every nonvanishing function preserver when
$\dim E =\dim F\lt +\infty$, is a weighted composition operator
We show that the map $\tau\colon Y\to X$ is a locally (little) Lipschitz homeomorphism.
(generalized, locally, little) Lipschitz functions, zero-set containment preservers, biseparating maps
46E40 - Spaces of vector- and operator-valued functions
54D60 - Realcompactness and realcompactification
46E15 - Banach spaces of continuous, differentiable or analytic functions