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  "body": "\n## Definition\n\nLet $X$ be a [[normed vector space|03J5RAKB30]], let $A \\subseteq X$ be nonempty, and let $x \\in X$.\nA point $a^* \\in A$ is a *best approximation* of $x$ from $A$ if\n$$\\norm{x - a^*} \\le \\norm{x - a} \\quad \\text{for all } a \\in A.$$\nEquivalently, $\\norm{x - a^*} = \\inf_{a \\in A} \\norm{x - a}$, an [[infimum|04KFTA34C5]] that is attained at $a^*$.\n\n## Remarks\n\nExample: in $\\R^2$ with the Euclidean norm, let $A$ be the horizontal axis and $x = (1, 2)$.\nFor $a = (t, 0)$ one has $\\norm{x - a}^2 = (1 - t)^2 + 4 \\ge 4$, with equality only at $t = 1$, so $(1, 0)$ is the best approximation.\n\nNon-example: in $\\R$, let $A = (0, 1]$ and $x = 0$.\nThe infimum of $\\abs{0 - a}$ over $A$ is $0$, but no $a \\in A$ attains it, so $x$ has no best approximation from $A$.\n\nOther names are nearest point and, in the uniform norm, minimax approximation.\nThe set of best approximations of $x$ from $A$ can be empty, a single point, or larger.\n",
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