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
- $k = \map {P_K} h \iff k \in K$ and $\map d {h, k} = \map d {h, K}$
where the latter $d$ signifies distance to a set.
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That $P_K$ is well-defined follows from Unique Point of Minimal Distance.
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
- Orthogonal (Hilbert Space), the origin of the nomenclature.
- Projection (Hilbert Spaces), an algebraic abstraction.
Sources
- 1990: John B. Conway: A Course in Functional Analysis ... (previous) ... (next) $I.2.8$
