An upper bound on the dimension of the reflexivity closure

Abstract

Let V, W be linear spaces over an algebraically closed field, and let S be an n―dimensional subspace of linear operators that maps V into W. We give a sharp upper bound for the dimension of the intersection of all images of nonzero operators from S, namely dim (∩ A∈S\{0} ImA ) ≤ dim V - n + 1. As an application, we also give a sharp upper bound for the dimension of the reflexivity closure Ref S of S, namely dim (RefS) ≤ n(n + 1)/2.