Minimal rank and reflexivity of operator spaces

Let ${\mathcal{S}}$ be an $n$-dimensional space of linear operators between the linear spaces $U$ and $V$ over an algebraically closed field $\mathbb{F}$. Improving results of Larson, Ding, and Li and Pan we show the following.

Theorem. Let $S_1, \ldots , S_n$ be a basis of $\mathcal{S}$. Assume that every nonzero operator in $\mathcal{S}$ has rank larger than $n$. Then a linear operator $T : U \to V$ belongs to $\mathcal{S}$ if and only if for every $u\in U$, $Tu$ is a linear combination of $S_1 u , \ldots , S_n u$.