In our functional analysis homework, we are asked to consider the norm sum/difference of two orthogonal projections $P_1, P_2$. For convenience, I am considering $P_1, P_2$ as orthogonal projections onto $M,N$ finite dimensional subspaces. This question is not difficult if we further assume $M,N$ are orthogonal to each other: let $\{\phi_n\},\{\psi_n\}_n$ be orthonormal bases for $M,N$ respectively. One can extend the union of these two collections to an orthonormal basis $\{\varphi_n\}$ of the Hilbert space $H$. Hence, we can write $$ P_1 \pm P_2 = \sum_{k=1}^\infty \lambda_k (\cdot, \varphi_k)\varphi_k. $$ Then one not only knows the norm of $P_1 \pm P_2$, but also knows even the Hilbert-Schmidt norm.
My question is: can we derive a more general statement if $M,N$ are not orthogonal to each other? I even got a bit stuck when I think about $M,N$ are just one-dimensional subspaces. Thanks a lot for the help in advance!