= Hilbert projection theorem
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= Closest point theorem in a Hilbert space
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For every nonempty <closed convex set> $C$ in a <Hilbert space> and every $x$ in that space, there is a unique $p\in C$ satisfying
$$
\|x-p\|=\inf_{z\in C}\|x-z\|.
$$
Closedness and completeness let a limiting approximation attain the infimum; <convexity> prevents two different minimizers. The proof idea is to choose a sequence approaching the infimum. The <parallelogram identity> and the admissible midpoint make it a <Cauchy sequence>; applying the same identity to two putative minimizers proves uniqueness.
For a <closed subspace of a Hilbert space>, this nearest-point map is an <orthogonal projection> and is <linear>. For a general <closed convex set> it need not be <linear>: projecting the real line onto $[0,1]$ clips values below zero to zero and values above one to one. The theorem is therefore useful for both linear approximation and constrained nearest-point problems.https://ocw.mit.edu/courses/18-102-introduction-to-functional-analysis-spring-2021/resources/18102-sp21-lecture-17/[Rodriguez, MIT 18.102, lecture 17: minimizers and orthogonal complements]{ref}
Closedness is essential even in familiar examples. For the open <interval> $(0,1)\subseteq\mathbb R$ and $x=0$, the distance infimum is zero but no nearest point exists. The same obstruction appears when approximating $(1/k)_{k\geq1}$ by <finitely supported sequences>: the infimum is zero, but the limit lies outside that subspace until we take its <Hilbert space completion>.
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