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

A Subspace of Maximal Dimension with Bounded Schmidt Rank