On the minimal rank in non-reflexive operator spaces over finite fields

Let U and V be vector spaces over a field K, and S be an n-dimensional linear subspace of L(U,V). The space S is called algebraically reflexive whenever it contains every linear map g:U→V such that, for all x∈U, there exists f∈S with g(x)=f(x). A theorem of Meshulam and Šemrl states that if S is not algebraically reflexive then it contains a non-zero operator of rank at most 2n−2, provided that K has more than n+2 elements. In the present article, we prove that the provision on the cardinality of the underlying field is unnecessary. To do so, we demonstrate that the above result holds for all finite fields.