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 map $P_K: H \to H$ 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 well-defined follows from Unique Point of Minimal Distance to Closed Convex Subset of Hilbert Space.

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

This and other properties of $P_K$ are collected in Properties of Orthogonal Projection.

Also see

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