Orthogonal Projection onto Orthocomplement

Theorem
Let $H$ be a Hilbert space.

Let $A$ be a closed linear subspace of $H$.

Let $P_A: H \to H$ be the orthogonal projection onto $A$.

Let $P_{A^\perp}: H \to H$ be the orthogonal projection onto $A^\perp$, the orthocomplement of $A$.

Let $I: H \to H$ be the identity operator on $H$.

Then:


 * $P_{A^\perp} = I - P_A$

Proof
Let $h \in H$.

By definition of orthogonal projection, $P_A h = a \in A$ such that $h - a \in A^\perp$.

Then the claim is that for all $h \in H$:


 * $\paren{ I - P_A } h = h - P_A h$

is in $A^\perp$, and:


 * $h - \paren{h - P_A h} \in \paren{ A^\perp }^\perp$

That $h - P_A h \in A^\perp$ is part of the definition of $P_A$.

For the other criterion, note:


 * $h - \paren{ h - P_A h } = P_A h \in A$

By Double Orthocomplement is Closed Linear Span: Corollary, $\paren{ A^\perp }^\perp = A$.