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) = ∅ ⇐⇒ z(φf) = ∅, ∀ 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, while our arguments are not usually seen in the literature.