Existence of optimal subspaces in reflexive Banach spaces

Abstract

Given a finite set $Y$ in a reflexive Banach space $F$ and a family $\mathcal{C}$ of closed subspaces of $F$, we study the problem of finding a subspace $V_0$ in $\mathcal{C}$ that best approximates the data $Y$ in the sense that $\sum_{f \in Y} d(f,V_0)= \min_{V \in \mathcal{C} }\sum_{f \in Y} d(f,V)$, where $d$ is the distance function on $F$. In this paper, we give necessary conditions and sufficient conditions over $\mathcal{C}$ for which such a best approximation exists. In particular, when $F$ has finite dimension a characterization on $\mathcal{C}$ is given.