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Let $A$ and $B$ be two closed subspaces of a given Hilbert space H. Let $h\in H$, $P(h|M)$ denotes the orthogonal projection of $h$ on the closed subspace $M$. Show that if $h\perp B$, then $P(h|A\vee B)=P(h|A)$, where $A\vee B$ is the closed the subspace spanned by A and B. Thanks.

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Hint: show that $\ h-P(h|A) \, \perp \, A\lor B$.

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