Properties of Orthogonal 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 $P_K$ has the following properties:


 * $(1):\qquad P_K$ is a linear transformation on $H$.
 * $(2):\qquad \forall h\in H: \left\|{P_K(h)}\right\| \le \left\|{h}\right\|$
 * $(3):\qquad P_K \circ P_K = P_K$
 * $(4):\qquad \ker P_K = K^{\perp}$ and $\operatorname{ran} P_K = K$