Given a projector $P$ of finite rank on Hilbert space $\mathcal{H}_{A}\otimes \mathcal{H}_{B}$, where $\mathcal{H}_{A}$ and $\mathcal{H}_{A}$ are infinite dimensional Hilbert spaces, is the schmidt decomposition of $P$ finite.
Intuitively it feels like this should be a yes, but I also know that infinite dimensions are something to be careful of. One thing that does come to mind is that for vector spaces (rather than bounded operators) we can construct $\sum_{i=1}^{\infty} c_{i} \vec{e}_{i}\otimes \vec{e}_{i}$ for clever choice of $c_{i}$. But the direct analogy when expanded to operators seems to not work (since the projector rank would be infinite).
I suspect this is a question with a well known answer but don't know where to look to find it.