# Definition:Orthogonal Projection

This page is about Orthogonal Projection in the context of Hilbert Spaces. For other uses, see Projection.

## Definition

Let $H$ be a Hilbert space.

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

Then the orthogonal projection on $K$ is the mapping $P_K: H \to H$ defined by

$k = \map {P_K} h \iff k \in K$ and $\map d {h, k} = \map d {h, K}$

where the latter $d$ signifies distance to a set.

That $P_K$ is indeed a mapping is proved on Orthogonal Projection is Mapping.

The name orthogonal projection stems from the fact that $\paren {h - \map {P_K} h} \perp K$.