The distance to the functions with range in a given set in Banach spaces of vector-valued continuous functions

In this paper we give a formula for the distance from an element f of the Banach space C(Omega,X)---where X is a Banach space and Omega is a compact topological space---to the subset C(Omega,S) of all functions whose range is contained in a given nonempty subset S of X. This formula is in terms of the norm in C(Omega) of the distance function to S that is induced by f (namely, of the scalar-valued function d_f^S which maps t \in Omega into the distance from f(t) to S), and generalizes the known property that the distance from f to C(Omega,V) be equal to the norm of d_f^V in C(Omega) for every vector subspace V of X ([Buc], Theorem 2; [FC], Lemma 2). Indeed, we prove that the distance from f to C(Omega,S) is larger than or equal to the norm of d_f^S in C(Omega) for every nonempty subset S of X, and coincides with it if S is convex or a certain quotient topological space of Omega is totally disconnected. Finally, suitable examples are constructed, showing how for each Omega such that the above mentioned quotient is not totally disconnected, the set S and the function f can be chosen so that the distance from f to C(Omega,S) be strictly larger than the C(Omega)-norm of d_f^S.