# Moreau's decomposition theorem

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- | + | '''Moreau's theorem''' is a fundamental result characterizing projections onto closed convex cones in Hilbert spaces. | |

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+ | Let <math>\mathcal K</math> be a closed convex cone in the Hilbert space <math>(\mathcal H,\langle\cdot,\cdot\rangle)</math> and <math>\mathcal K^\circ</math> its polar. Denote by <math>P_{\mathcal K}</math> and <math>P_{\mathcal K^\circ}</math> the projections onto <math>\mathcal K</math> and <math>\mathcal K^\circ</math>, respectively. For <math>x,y,z\in\mathcal H</math> the following two statements are equivalent: | ||

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+ | <ol> | ||

+ | <li><math>z=x+y</math>, <math>x\in\mathcal K, y\in\mathcal K^\circ</math> and <math>\langle x,y\rangle=0</math></li> | ||

+ | <li><math>x=P_{\mathcal K}z</math> and <math>y=P_{\mathcal K^\circ}z</math> | ||

+ | </li> | ||

+ | </ol> |

## Revision as of 14:50, 10 July 2009

**Moreau's theorem** is a fundamental result characterizing projections onto closed convex cones in Hilbert spaces.

Let be a closed convex cone in the Hilbert space and its polar. Denote by and the projections onto and , respectively. For the following two statements are equivalent:

- , and
- and