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  "body": "\n## Summary\n\nThe error of best approximation of $x$ from a set $A$ is the distance from $x$ to $A$, the infimum of $\\norm{x - a}$ over $a \\in A$.\nFor a function $f$ and the polynomials of degree at most $n$ it is written $E_n(f)$.\nIt measures how well $f$ can be approximated at all, whether or not a [[best approximation|7N4988HN3P]] exists.\nIts decay as $n \\to \\infty$ reflects the smoothness of $f$.\n\n## Explanations\n\n### L1 human\n\nIt is the smallest gap you can achieve with the tools you are allowed to use.\nFor $f(x) = x^2$ on $[0, 1]$ and uniform distance, the best constant is $1/2$ with gap $1/2$, the best line has gap $1/8$, and a quadratic gives gap $0$.\nAllowing more complicated tools never makes the gap larger, so $E_0(f) \\ge E_1(f) \\ge E_2(f) \\ge \\cdots$.\nFor a continuous function on a closed interval these numbers tend to $0$.\n",
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