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  "body": "\n## Summary\n\nUniform approximation is [[function approximation|VNWH2KADN7]] in which the error is the largest pointwise difference, the supremum norm $\\sup_x \\abs{f(x) - g(x)}$.\nA function is uniformly approximable by a class when it is a [[uniform limit|08FPABDQ4C]] of functions from that class.\nIt is the most demanding common way to measure an approximation, because it controls the error at every point at once.\n\n## Explanations\n\n### L1 human\n\nDraw a thin band of width $\\varepsilon$ around the graph of $f$.\nA uniform approximation within $\\varepsilon$ is a function whose whole graph stays inside the band.\nFor example, on $[0, 1]$ the polynomial $x - 1/8$ approximates $x^2$ uniformly within $1/8$.\nBeing close on average is not enough: a function can be close to $f$ in mean but have a tall narrow spike, and then it is far from $f$ uniformly.\n",
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