Kernel of Orthogonal Projection on Closed Linear Subspace of Hilbert Space

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:


 * $\ker P_K = K^\bot$

where:
 * $\ker P_K$ denotes the kernel of $P_K$
 * $K^\bot$ denotes the orthocomplement of $K$.

Proof
We first prove that:


 * $\ker P_K \subseteq K^\bot$

Let $h \in \ker P_K$.

Then:


 * $\map {P_K} h = 0$

From Unique Point of Minimal Distance to Closed Convex Subset of Hilbert Space, ​we have:


 * $h - \map {P_K} h \in K^\bot$

That is:


 * $h \in K^\bot$

So:


 * $\ker P_K \subseteq K^\bot$

We now prove that:


 * $K^\bot \subseteq \ker P_K$

Let $h \in K^\bot$.

From Unique Point of Minimal Distance to Closed Convex Subset of Hilbert Space, we have that:


 * $\map {P_K} h$ is the unique point such that $h - \map {P_K} h \in K^\bot$

So, since $h - 0 \in K^\bot$, we have $\map {P_K} h = 0$.

So, we have $h \in \ker P_K$, and we obtain:


 * $K^\bot = \ker P_K$