Chebyshev Centers of Compact Sets with Respect to Stone-Weierstrass Subspaces

Abstract

Publisher Summary This chapter focuses on Chebyshev centers of compact sets with respect to Stone–Weierstrass subspaces. Various theorems and proofs are presented in the chapter in this regard. A formula for the relative Chebyshev radius in terms of the Chebyshev radius of the corresponding set valued map is presented in the chapter. The proximinality of all Stone–Weierstrass subspaces implies the existence of relative Chebyshev centers for all compact subsets of C(S,X) . C(S,X) is the Banach space of all continuous functions on a compact Hausdorff space (S) with values in a Banach space (X) equipped with the supremum norm. Any Stone–Weierstrass subspace is proximinal if X is the real line (a subspace G of a normed linear space Y is called proximinal if every y ∈ Y possesses an element of best approximation x in G) . A subspace V of C(S,X) is considered to be a Stone–Weierstrass subspace of C(S,X) if there is a compact Hausdorff space T and a continuous surjection.