Best approximation of bounded functions by continuous functions

Abstract

It is a classical result that every closed subalgebra of C(X: ;,‘). X compact. is a proximinal subspace. Proofs of this result were published by Pelczynski and Semadeni. but the result itself was known to Mazur. Several generalizations and extensions of Mazur’s theorem have appeared in the literature. For example. one may consider the problem of existence of Chebyshev centers for bounded subsets of C(X: !:c), i.e., the problem of deciding when C(X: ‘1: ) admits centers, This was settled by Kadets and Zamyatin, for X = ICI, b]. and by Garkavi, for X compact (see ) 13 / and 1121). The existence of relative Chebyshev centers (also called restricted centers) with respect to a closed subalgebra if c C(X . ’ 4;) was established in 1975 by Smith and Ward (see /24/J for any compact space X. An extension of this result to bounded functions, i.e., to closed subalgebras of I, (X: ; f was obtained by Mach in 1979 (see / IS/). Another line of generalizations of Mazur‘s theorem consists of the consideration of vector-valued functions. The problem of the existence of Chebyshev centers for bounded subsets of C,(X: E), the space of continuous and bounded E-valued functions, was solved by Ward in 1974 (see 125 /) in the following two cases: (a) E is a finite-dimensional strictly convex normed space and X is paracompact; (b) E is a Hilbert space and X is normal. Amir in 1978 (see 111) generalized both results by proving that C,,(X: E) admits centers when E is a uniformly convex Banach space and X is any topological space. When WC C,(X; E) is a closed vector subspace one asks whether it is proximinai or, more generally. whether any bounded subset of C,(X;E) has a relative Chebyshev center with respect to 6V. Along this line we have the study of proximinality of Grothendieck subspaces (in particular Stone13.5 OOZI-9045184 $3.00