Sum of Projections
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Theorem
Binary Case
Let $H$ be a Hilbert space.
Let $P, Q$ be projections.
Then $P + Q$ is a projection if and only if $\Rng P \perp \Rng Q$.
General Case
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
Sources
- 1990: John B. Conway: A Course in Functional Analysis (2nd ed.) ... (previous) ... (next): $\text {II}.3$ Exercises $4, 5$
- 1990: John B. Conway: A Course in Functional Analysis (2nd ed.) $\text {II}.7.1$