Let $\mathcal{H}$ be a Hilbert space and let $P$ and $Q$ be two orthogonal projections to closed subspaces $M$ and $N$ respectively.
Prove that:
- If $PQ$ is an orthogonal projection then it's range is $M\cap N$
- $PQ$ is orthogonal iff $PQ = QP$
I got stuck with the first clause, any hint would be most welcomed.