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 ( ϕ f ) = 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.