Fixed Points of Orthogonal Projection

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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 $h \in H$.


$\map {P_K} h = h$

if and only if $h \in K$.


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:

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

if and only if:

$h = \map {P_K} h$

So, we have:

$h = \map {P_K} h$

if and only if:

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

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

$h \in K$