Existence of “Larger” Projection?

Question

Suppose $H$ is a separable Hilbert space. If $P,Q$ are two finite dimension orthonormal projection, is there a finite dimension orthonormal projection $L$, st:

$$ L(H)\supset P(H), L(H)\supset Q(H)? $$

Answer 1

Take $L$ to be the orthogonal projection onto the finite dimensional subspace $P(H)+Q(H)$.