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  "body": "\n## Statement\n\nLet $K$ be a [[compact|0TVJZYB6SH]] [[Hausdorff space|0S1FJ8VEX9]], and let $C(K)$ be the space of continuous functions $K \\to \\R$ with the supremum norm.\nLet $V \\subseteq C(K)$ be a [[linear subspace|00YC1D652K]], let $f \\in C(K)$ and $p^* \\in V$, and let\n$$E = \\set{x \\in K : \\abs{f(x) - p^*(x)} = \\norm{f - p^*}}$$\nbe the set of extreme points of the error.\nThen $p^*$ is a [[best approximation|7N4988HN3P]] of $f$ from $V$ if and only if\n$$\\min_{x \\in E} \\big( f(x) - p^*(x) \\big) \\, p(x) \\le 0 \\quad \\text{for every } p \\in V.$$\n\n## Remarks\n\nThe idea of the proof: suppose some $p \\in V$ has $(f - p^*) p \u003e 0$ on $E$.\nBy compactness this product is positive on an open set $U \\supseteq E$, and $\\abs{f - p^*}$ stays below $\\norm{f - p^*} - \\eta$ for some $\\eta \u003e 0$ off $U$.\nThen $\\norm{f - p^* - t p} \u003c \\norm{f - p^*}$ for small $t \u003e 0$, so $p^*$ is not best.\nConversely, if $q \\in V$ has $\\norm{f - q} \u003c \\norm{f - p^*}$, then $p = q - p^*$ satisfies $(f - p^*) p = (f - p^*)^2 - (f - p^*)(f - q) \u003e 0$ on $E$.\n\nExample: for $f(x) = x^2$ on $[0, 1]$, $V$ the lines and $p^*(x) = x - 1/8$, the set $E$ is $\\set{0, 1/2, 1}$ with error signs $+, -, +$.\nA line $p$ with $p(0) \u003e 0$, $p(1/2) \u003c 0$ and $p(1) \u003e 0$ would have two zeros, so it does not exist, and the criterion shows $p^*$ is best.\n\nFor complex-valued functions the same criterion holds with $\\operatorname{Re} \\big( \\overline{(f(x) - p^*(x))} \\, p(x) \\big)$ in place of the product.\n\n## Explanations\n\n### L1 human\n\nA candidate approximation is best unless some correction pushes in the right direction at every point where the error is worst.\nThe criterion checks exactly this: there must be no $p$ that has the same sign as the error at all the worst points.\n\n### L2 human\n\nThe criterion holds for every linear subspace, not only for polynomials or Haar subspaces.\nFor Haar subspaces it turns into the alternation property of the equioscillation theorem, because a function of a Haar subspace can follow any sign pattern with few sign changes.\nIt is a form of the first-order optimality condition for the convex function $p \\mapsto \\norm{f - p}$.\nSimilar criteria hold in other normed spaces with the norm's subdifferential in place of the product.\n",
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