Sum of Projections/General Case

Theorem
Let $H$ be a Hilbert space.

Let $\left({M_i}\right)_{i \in I}$ be an $I$-indexed set of closed linear subspaces of $H$.

Let $M_i$ and $M_j$ be orthogonal whenever $i \ne j$.

Denote, for each $i \in I$, by $P_i$ the orthogonal projection onto $M_i$.

Denote by $P$ the orthogonal projection onto the closed linear span $\vee \left\{{M_i: i \in I}\right\}$ of the $M_i$.

Then for all $h \in H$, $\displaystyle \sum \left\{{P_i h: i \in I}\right\} = Ph$, where $\displaystyle \sum$ denotes a generalized sum.

Also see

 * Product of Projections
 * Difference of Projections