Orthogonal Projection is Projection

Theorem
Let $H$ be a Hilbert space.

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

Let $P_K$ denote the orthogonal projection on $K$.

Then $P_K$ is idempotent, i.e.:


 * $P_K \circ P_K = P_K$

Proof
Let $h \in H$.

From the definition of the orthogonal projection, we have:


 * $\map {P_K} h \in K$

So, from Fixed Points of Orthogonal Projection, we have:


 * $\map {\paren {P_K \circ P_K} } h = \map {P_K} {\map {P_K} h} = \map {P_K} h$

Since $h$ was arbitrary, we have:


 * $P_K \circ P_K = P_K$