Definition:Orthogonal Projection

This page is about orthogonal projections in Hilbert spaces. For other uses, see Definition: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.

Sources

• John B. Conway: A Course in Functional Analysis (1990)... (previous)... (next) $I.2.8$