: Mathématiques: Hilbert projection theorem

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Hilbert projection theorem

In mathematics, the Hilbert projection theorem is a famous result of convex analysis that says that for every vector ${\\displaystyle x}$ in a Hilbert space ${\\displaystyle H}$ and every nonempty closed convex ${\\displaystyle C\\subseteq H,}$ there exists a unique vector ${\\displaystyle m\\in C}$ for which ${\\displaystyle \\|c-x\\|}$ is minimized over the vectors ${\\displaystyle c\\in C}$ ; that is, such that ${\\displaystyle \\|m-x\\|\\leq \\|c-x\\|}$ for every ${\\displaystyle c\\in C.}$