Hilbert projection theorem

From Wikipedia, the free encyclopedia

The Hilbert Projection Theorem is a famous result of convex analysis that says that for every point $x$ in a Hilbert space $H$ and every closed subspace $M \subset H$ , there exists a unique point $m \in M$ for which $\lVert x - m \rVert$ is minimized over $M$ . A necessary and sufficient condition for $m$ is that the vector $x - m$ be orthogonal to $M$ .