Suppose $H$ is a Hilbert space and there are closed convex sets $C,D \subset H$ such that every $h \in H$ can be written as $h= c + d$ where $c \in C$ and $d \in D$.
Denote by $P_C$ the orthogonal projection onto the subset $C$. It is not true that $$h=P_{C}(h) + P_{D}(h)$$ in general, is it? Are there some conditions under which it is true?