The embedding of proximinal sets

Abstract

A subset M of a normed linear space X is said to be proximinal if inf m ϵ M ∥ x − m∥ is attained, for each x ϵ X. If X is embedded in another normed space, Z, a proximinal subset of X may or may not be proximinal in Z. Certain practical problems in multivariate approximation lead us to examine the case when X = C( S) and Z = C( S × T), where S and T are compact Hausdorff spaces. We characterize the proximinal subspaces of C( S) which are proximinal in C( S × T) for every T. In another section, a generalization of Mazur's Proximinality Theorem is given. This generalization gives a condition under which a subspace of functions ν ° f is proximinal, when f is held fixed and ν ranges over a proximinal subspace.