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(@W,X)-where X is a Banach space and @W is a compact topological space-to the subset C(@W,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(@W) of the distance function to S that is induced by f (namely, of the scalar-valued function d"f^S which maps t@?@W into the distance from f(t) to S), and generalizes the known property that the distance from f to C(@W,V) be equal to the norm of d"f^V in C(@W) 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(@W,S) is larger than or equal to the norm of d"f^S in C(@W) for every nonempty subset S of X, and coincides with it if S is convex or a certain quotient topological space of @W is totally disconnected. Finally, suitable examples are constructed, showing how for each @W, 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(@W,S) be strictly larger than the C(@W)-norm of d"f^S.