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  "body": "\n## Definition\n\nLet $X$ be a [[locally compact|0FE2SB8WVJ]] [[Hausdorff space|0S1FJ8VEX9]] and let $\\mathbb{K}$ be $\\R$ or $\\C$.\nA [[continuous function|0X3WGX95FE]] $f \\colon X \\to \\mathbb{K}$ *vanishes at infinity* if, for every $\\varepsilon \u003e 0$, the set $\\set{x \\in X : \\abs{f(x)} \\ge \\varepsilon}$ is [[compact|0TVJZYB6SH]].\nThe *space of functions vanishing at infinity* $C_0(X)$ is the set of these functions, with pointwise operations and the norm $\\norm{f}_\\infty = \\sup_{x \\in X} \\abs{f(x)}$.\n\n## Remarks\n\nExample: $f(x) = e^{-x^2}$ on $\\R$.\nFor $0 \u003c \\varepsilon \\le 1$, the set $\\set{\\abs{f} \\ge \\varepsilon}$ is the closed bounded interval $\\abs{x} \\le \\sqrt{\\log(1/\\varepsilon)}$, which is compact.\nFor $\\varepsilon \u003e 1$ it is empty.\n\nNon-example: the constant function $1$ on $\\R$.\nFor $\\varepsilon = 1/2$, the set $\\set{\\abs{f} \\ge 1/2}$ is all of $\\R$, which is not compact.\n\nEach $f \\in C_0(X)$ is bounded: $\\abs{f} \u003c 1$ off the compact set $\\set{\\abs{f} \\ge 1}$, and $\\abs{f}$ attains a maximum on that set if it is nonempty.\nSo $C_0(X)$ is a subspace of the [[space of bounded continuous functions|MG3X8Y1VT4]], and it is closed there, so it is a Banach space.\nIf $X$ is compact, then $C_0(X) = C(X)$.\nFor $X = \\N$ with the discrete topology, $C_0(\\N)$ is the sequence space $c_0$.\nThe compactly supported continuous functions are dense in $C_0(X)$.\n",
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