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

Abstract

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 d f S 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 d f V 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 d f S 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 d f S .