Existence of optimal subspaces in reflexive Banach spaces

Abstract

Given a finite set Y in a reflexive Banach space F and a family C of closed subspaces of F, we study the problem of finding a subspace W in C that best approximates the data Y in the sense that sum_{f in Y} d(f,W) = min_{V in C} sum_{f in Y} d(f,V), where d is the distance function on F. In this paper, we give necessary conditions and sufficient conditions over C for which such a best approximation exists. In particular, when F has finite dimension a characterization on C is given.