Definition:Orthogonal Projection

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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.


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


Sources