# Definition:Orthogonal Projection

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*This page is about Orthogonal Projection in the context of Hilbert Space. 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 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 to Closed Convex Subset of Hilbert Space.

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

- Definition:Orthogonal (Hilbert Space)], the origin of the nomenclature.
- Definition:Projection (Hilbert Spaces), an algebraic abstraction.

## Sources

- 1990: John B. Conway:
*A Course in Functional Analysis*... (previous) ... (next) $\text I.2.8$

- 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**orthogonal projection**