Real and complex analysis, 3rd ed.
Real and complex analysis, 3rd ed.
Approximation with monotone norms in tensor product spaces
Journal of Approximation Theory
Approximation by extreme functions
Journal of Approximation Theory
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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.