# Moreau's decomposition theorem

(Difference between revisions)
 Revision as of 19:11, 8 July 2009 (edit)← Previous diff Revision as of 14:50, 10 July 2009 (edit) (undo)Next diff → Line 1: Line 1: - 1962 + '''Moreau's theorem''' is a fundamental result characterizing projections onto closed convex cones in Hilbert spaces. + + Let $\mathcal K$ be a closed convex cone in the Hilbert space $(\mathcal H,\langle\cdot,\cdot\rangle)$ and $\mathcal K^\circ$ its polar. Denote by $P_{\mathcal K}$ and $P_{\mathcal K^\circ}$ the projections onto $\mathcal K$ and $\mathcal K^\circ$, respectively. For $x,y,z\in\mathcal H$ the following two statements are equivalent: + +
+
1. $z=x+y$, $x\in\mathcal K, y\in\mathcal K^\circ$ and $\langle x,y\rangle=0$
2. +
3. $x=P_{\mathcal K}z$ and $y=P_{\mathcal K^\circ}z$ +
4. +

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

1. $LaTeX: z=x+y$, $LaTeX: x\in\mathcal K, y\in\mathcal K^\circ$ and $LaTeX: \langle x,y\rangle=0$
2. $LaTeX: x=P_{\mathcal K}z$ and $LaTeX: y=P_{\mathcal K^\circ}z$