  We study Schmidt rank for a vector (i.e., a pure state) and Schmidt number
for a mixed state which are entanglement measures. We show that if a subspace
of a certain bipartite system contains no vector of Schmidt rank $\leqslant k$,
then any state supported on that space has Schmidt number at least $k+1$. A
construction of subspace of $\mathbb{C}^m \otimes \mathbb{C}^n$ of maximal
dimension, which does not contain any vector of Schmidt rank less than $3$, is
given here.
