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(Ω,X)-where X is a Banach space and Ω is a compact topological space-to the subset C(Ω,S) of all functions whose range is contained in a given nonempty subset S of X. This formula is given in terms of the norm in C(Ω) of the distance function to S that is induced by f (namely, of the scalar-valued function dfS which maps t∈Ω into the distance from f(t) to S), and generalizes the known property that the distance from f to C(Ω,V) be equal to the norm of dfV in C(Ω) for every vector subspace V of X [Buck, Pacific J. Math. 53 (1974) 85-94, Theorem 2; Franchetti and Cheney, Boll. Un. Mat. Ital. B (5) 18 (1981) 1003-1015, Lemma 2]. Indeed, we prove that the distance from f to C(Ω,S) is larger than or equal to the norm of dfS in C(Ω) for every nonempty subset S of X, and coincides with it if S is convex or a certain quotient topological space of Ω is totally disconnected. Finally, suitable examples are constructed, showing how for each Ω, 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(Ω,S) be strictly larger than the C(Ω)-norm of dfS.