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  "body": "\n## Statement\n\nLet $H$ be a real or complex [[Hilbert space|HM2HHE3J51]].\nLet $C$ be a nonempty [[closed|0ZCK8DQYN1]] [[convex|02AS45T3H9]] subset of $H$, and let $x \\in H$.\nThen there is exactly one $y \\in C$ such that\n$$\\norm{x - y} = \\inf_{z \\in C} \\norm{x - z}.$$\nThis point $y$ is the only point of $C$ with $\\operatorname{Re} \\ip{x - y}{z - y} \\le 0$ for all $z \\in C$.\n\n## Remarks\n\nThe idea of the proof: let $d$ be the infimum and take $z_n \\in C$ with $\\norm{x - z_n} \\to d$.\nThe [[parallelogram law|026HPGYGS4]] gives $\\norm{z_n - z_m}^2 = 2 \\norm{x - z_n}^2 + 2 \\norm{x - z_m}^2 - 4 \\norm{x - (z_n + z_m)/2}^2$.\nBy convexity the midpoint lies in $C$, so the last norm is at least $d$, and $(z_n)$ is a Cauchy sequence.\nCompleteness and closedness give a limit $y \\in C$, and the same identity gives uniqueness.\n\nWhen $C$ is a closed linear subspace $M$, the condition becomes $x - y \\perp M$.\nThen $y$ is the [[orthogonal projection|0B8KZRT28M]] of $x$ on $M$, and $H = M \\oplus M^\\perp$ with the [[orthogonal complement|0XJXTBCB2T]] $M^\\perp$.\nFor finite-dimensional $M$ this is the [[closest point property|0MCDDSCB9Z]].\n\nThe hypotheses matter.\nIf $C$ is not closed, such as an open ball not containing $x$, there may be no nearest point.\nIf $C$ is not convex, such as a circle in $\\R^2$ and $x$ its center, the nearest point need not be unique.\nIn the Banach space $\\R^2$ with the max norm, the nearest point of the line $\\set{(t, 1)}$ to $0$ is not unique.\n\n## Explanations\n\n### L1 human\n\nFrom any point there is exactly one nearest point in a closed convex set.\nThe segment from the point to its nearest point meets the set at a right angle or wider.\nThis is what makes orthogonal projection work in infinite dimensions.\n\n### L2 human\n\nThe theorem is the base of Hilbert space geometry.\nIt gives the orthogonal decomposition $H = M \\oplus M^\\perp$, the existence of orthogonal projections and the Riesz representation theorem.\nIn optimisation and approximation it shows that least squares problems over closed convex sets have unique solutions.\n",
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