As a by-product of a numerical linear algebra result on structured matrices, I can prove the following result:

For each $m$-dimensional subspace $\mathcal{U}$ of $\mathbb{C}^n$, one can find a $n\times n$ permutation matrix $\Pi$ such that $\mathcal{U}=\operatorname{span} \Pi\begin{bmatrix}I\\\\X\end{bmatrix}$ and all the entries of $X$ satisfy $\left\vert X_{ij}\right\vert\leq 1$.

A self-contained proof is not hard.

The maps $X\mapsto \Pi \begin{bmatrix}I\\\\X\end{bmatrix}$ are local charts that form perhaps the simplest atlas of the $(n,m)$ Grassmannian, so I imagine this could be a natural question for people studying Grassmannians. I'd be surprised if this result were not already known.

Do you have a reference for it? Seeing a new theory and a different approach that produces the same result could give me new insight on the original problem I am studying.