Fixed Points of Orthogonal Projection

Theorem
Let $\struct {H, \innerprod \cdot \cdot}$ be a Hilbert space.

Let $\norm \cdot$ be the inner product norm of $H$.

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

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

Let $h \in H$.

Then:


 * $\map {P_K} h = h$

$h \in K$.

Proof
Let $d$ be the metric induced by $\norm \cdot$.

Let $h \in H$.

By the definition of orthogonal projection, we have:


 * $\map d {h, \map {P_K} h} = \map d {h, K}$

Note that by the definition of a metric, we have that:


 * $\map d {h, \map {P_K} h} = 0$




 * $h = \map {P_K} h$

So, we have:


 * $h = \map {P_K} h$




 * $\map d {h, K} = 0$

Since $K$ is closed, from Subset of Metric Space is Closed iff contains all Zero Distance Points, this is the case :


 * $h \in K$