Best approximation in LP (I, X)

Abstract

Let T be a measure space and m a finite measure on T. The space of p-Bochner integrable functions defined on T with values in a Banach space X is denoted by Lp(T, X). It is well known (1) that Lp (T, X) is a Banach space under the normA subspace E in a Banachh space F is said to be proximinal if for each xF there is at least one xE such thatThe element y is called best approximant of x in E.