Hilbert projection theorem

In mathematics, the Hilbert projection theorem is a famous result of convex analysis that says that for every vector x {\displaystyle x} in a Hilbert space H {\displaystyle H} and every nonempty closed convex C ⊆ H , {\displaystyle C\subseteq H,} there exists a unique vector m ∈ C {\displaystyle m\in C} for which ‖ c − x ‖ {\displaystyle \|c-x\|} is minimized over the vectors c ∈ C {\displaystyle c\in C} ; that is, such that ‖ m − x ‖ ≤ ‖ c − x ‖ {\displaystyle \|m-x\|\leq \|c-x\|} for every c ∈ C . {\displaystyle c\in C.} == Finite dimensional case == Some intuition for the theorem can be obtained by considering the first order condition of the optimization problem.

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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 x {\displaystyle x} in a Hilbert space H {\displaystyle H} and every nonempty closed convex C ⊆ H , {\displaystyle C\subseteq H,} there exists a unique vector m ∈ C {\displaystyle m\in C} for which ‖ c − x ‖ {\displaystyle \|c-x\|} is minimized over the vectors c ∈ C {\displaystyle c\in C} ; that is, such that ‖ m − x ‖ ≤ ‖ c − x ‖ {\displaystyle \|m-x\|\leq \|c-x\|} for every c ∈ C . {\displaystyle c\in C.} == Finite dimensional case == Some intuition for the theorem can be obtained by considering the first order condition of the optimization problem.

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Source: Wikipedia "Hilbert projection theorem" · CC BY-SA 4.0

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