Sum of Projections/General Case

Theorem
Let $H$ be a Hilbert space.

Let $\family {M_i}_{i \mathop \in I}$ 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 $\vee \set {M_i: i \in I}$ of the $M_i$.

Then for all $h \in H$:
 * $\ds \sum \set {P_i h: i \in I} = P h$

where $\ds \sum$ denotes a generalized sum.

Also see

 * Product of Projections
 * Difference of Projections