Definition:Orthogonal 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 $\paren {h - \map {P_K} h} \perp K$.

Also see

 * Definition:Orthogonal (Linear Algebra), the origin of the nomenclature.
 * Definition:Projection (Hilbert Spaces), an algebraic abstraction.


 * Properties of Orthogonal Projection