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Let $H$ be a Hilbert space. Suppose that the linear subspace $\Lambda$ can be expressed as the orthogonal direct sum of the linear subspace $\Lambda_1$ with the one-dimensional linear subspace $\Lambda_2$. Does it follow that $\text{Proj}_{\Lambda} v = \text{Proj}_{\Lambda_1} v + \text{Proj}_{\Lambda_2} v$ for any $v \in H$?

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up vote 2 down vote accepted

Yes, and you don't need $\Lambda_2$ to be one-dimensional for this to work; anytime $P$ and $Q$ are projections onto orthogonal subspaces, $P + Q$ is projection onto the sum of those subspaces.

(By the way, I'm assuming all subspaces in question are closed, as projections don't work well otherwise.)

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