CONTINUITY OF THE ONE-SIDED BEST UNIFORM APPROXIMATION OPERATOR

Abstract

The purpose of this paper is to relate the continuity and selection properties of the one-sided best uniform approximation operator to similar properties of the metric projection. Let M be a closed subspace of C(T) which contains constants. Then the one-sided best uniform approximation operator is Hausdorff continuous (resp. Lipschitz continuous) on C(T) if and only if the metric projection P_M is Haudorff continuous (resp. Lipschitz continuous) on C(T). Also, the metric projection P_M admits a continuous (resp. Lipschitz continuous) selection if and only if the one-sided best uniform approximation operator admits a continuous (resp. Lipschitz continuous) selection.