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  "body": "\n## Statement\n\nLet $V$ be an inner product space over $\\R$ or $\\C$, let $W$ be a finite-dimensional subspace of $V$, and let $v \\in V$.\nLet $P_W$ be the [[orthogonal projection|0B8KZRT28M]] onto $W$.\nThen, for all $w \\in W$,\n$$\\norm{v - P_W(v)} \\le \\norm{v - w},$$\nwith equality only for $w = P_W(v)$.\n\n## Remarks\n\nThe proof: $v - w = (v - P_W(v)) + (P_W(v) - w)$ is a sum of orthogonal vectors, so the Pythagorean theorem applies.\nThe number $\\norm{v - P_W(v)}$ is the distance from $v$ to $W$.\nThe Hilbert projection theorem gives the same result for closed convex sets in Hilbert spaces.\nThis is the basis of the method of [[least squares|0BYTQ4X1HC]].\n\n## Explanations\n\n### L1 human\n\nThe nearest point of a subspace to a given vector is the foot of the perpendicular.\nEvery other point of the subspace is farther away, by the Pythagorean theorem.\nThis turns problems of best approximation into problems of projection.\n",
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