Let X be a Banach Space and $L, M$ closed subspaces such that $L\cap M = \{0\}$. I would like to apply Banach Isomorphism Theorem to prove that if $L+M$ is closed, then $P : L + M \to L$ defined as $P(l+m) = l$ is continuous. Any hint, please?
Also, for the converse, I don't now how to avoid the fact that $\cup_l P^{-1}(l)$ is not necessarily closed since it is a non-numerable union...