Kernel 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:


 * $\ker P_K = K^\bot$

and:


 * $\operatorname {ran} P_K = K$

where:
 * $\ker P_K$ denotes the kernel of $P_K$
 * $K^\bot$ denotes the orthocomplement of $K$
 * $\operatorname {ran} P_K$ denotes the image of $P_K$.