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  "body": "\n## Definition\n\nLet $S$ be a set and let $X$ be a [[normed vector space|03J5RAKB30]] of [[functions|0EPAT51DXQ]] on $S$.\nLet $A \\subseteq X$ be a nonempty subset, the approximating class.\nLet $f \\in X$ and $\\varepsilon \u003e 0$.\nAn *approximation of $f$ from $A$ within $\\varepsilon$* is a function $g \\in A$ with $\\norm{f - g} \\le \\varepsilon$.\nThe function $f$ *can be approximated arbitrarily well from $A$* if such a $g$ exists for every $\\varepsilon \u003e 0$.\nEquivalently, $f$ lies in the closure of $A$.\n\n## Remarks\n\nExample: take $X = C[0, 1]$ with the supremum norm, $A$ the polynomials of degree at most $2$, and $f(x) = e^x$.\nThe function $g(x) = 1 + x + x^2/2$ is an approximation within $0.22$.\nIndeed $e^x - g(x)$ is increasing on $[0, 1]$, since its derivative $e^x - 1 - x$ is nonnegative, so its maximum is $e - 2.5 \u003c 0.22$.\n\nNon-example: on $[-1, 1]$ with the supremum norm, the sign function cannot be approximated within any $\\varepsilon \u003c 1$ by continuous functions.\nLet $g$ be continuous with $g(0) = c$.\nNear $0$ the gaps $\\abs{1 - g(x)}$ for $x \u003e 0$ and $\\abs{-1 - g(x)}$ for $x \u003c 0$ come close to $\\abs{1 - c}$ and $\\abs{1 + c}$.\nOne of these two numbers is at least $1$.\n\nThe choice of norm matters: the sign function can be approximated arbitrarily well by continuous functions in the $L^2$ norm.\n",
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