Approximating $AB$ by $AP$ where $P$ is an orthogonal projection

Question

Let $A\in {R} ^{m \times d}$, $B \in R^{d \times d}$. Given $n\leq d$, I want to find an orthogonal projection $P$ onto an $n$-dimensional subspace of $R^d$ s.t $AP$ best approximates $AB$ according to some reasonable measure of approximation. Does this problem have a closed form solution?