Composition of orthogonal projections

Question

I need to prove the following result: Suppose $P_1$ and $P_2$ are orthogonal projections onto closed subspaces $V_1$ and $V_2$, then $P_1P_2x = x$ if and only if $x\in V_1\cap V_2$. But it seems to me that if you take two parallel lines $V_1$, $V_2$ in $\mathbb R^2$, and $x\in V_1$, then $P_1P_2x =x$, but $x\not\in V_1\cap V_2$. Can you please tell me what is wrong with my intuition?

Thanks!

If $V_1$ and $V_2$ are distinct parallel lines, then at least one of them does not represent a subspace of $\Bbb R^2$ (as one doesn't pass through the origin). — David Mitra, Feb 10 '13 at 16:54

score 1 · Accepted Answer · answered Feb 10 '13 at 18:18

"Subspace" in the appropriate context probably means linear subspace, closed under addition and closed under scalar multiplication. That implies they pass through the origin. Two parallel lines cannot both pass through the origin, so they're not both "subspaces" in the relevant sense.

Composition of orthogonal projections

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