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


 * $k = P_K(h) \iff k \in K$ and $d \left({h, k}\right) = d \left({h, K}\right)$

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

That $P_K$ is well-defined follows from Unique Point of Minimal Distance.

The name orthogonal projection stems from the fact that $\left({h - P_K \left({h}\right)}\right) \perp K$.

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

Also see

 * Orthogonal (Hilbert Space), the origin of the nomenclature.
 * Projection (Hilbert Spaces), an algebraic abstraction.