Definition:Orthogonal Projection

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This page is about orthogonal projection in the context of Hilbert spaces. For other uses, see 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.


This article is complete as far as it goes, but it could do with expansion.
In particular: Include definition $Ph = k$ s.t. $h - k \in K^\perp$
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This page or section has statements made on it that ought to be extracted and proved in a Theorem page.
In particular: There is no mention of this fact on Properties of Orthogonal Projection
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The name orthogonal projection stems from the fact that $\paren {h - \map {P_K} h} \perp K$.


Also see

  • Results about orthogonal projections can be found here.


Sources


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