Banach-Stone theorems for vector valued functions on completely regular spaces

We obtain several Banach-Stone type theorems for vector-valued functions in this paper. Let X,Y be realcompact or metric spaces, E,F locally convex spaces, and ϕ a bijective linear map from C(X,E) onto C(Y,F). If ϕ preserves zero set containments, i.e., z(f)⊆z(g)⟺z(ϕ(f))⊆z(ϕ(g)),∀f,g∈C(X,E), then X is homeomorphic to Y, and ϕ is a weighted composition operator. The above conclusion also holds if we assume a seemingly weaker condition that ϕ preserves nonvanishing functions, i.e., z(f)=0̸⟺z(ϕf)=0̸,∀f∈C(X,E). These two results are special cases of the theorems in a very general setting in this paper, covering bounded continuous vector-valued functions on general completely regular spaces, and uniformly continuous vector-valued functions on metric spaces. Our results extend and generalize many recent ones.