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  "body": "\n## Definition\n\nLet $X$ be a [[topological space|0CQHFPGJ45]] and let $\\mathbb{K}$ be $\\R$ or $\\C$.\nThe *space of bounded continuous functions* $C_b(X)$ is the set of [[continuous functions|0X3WGX95FE]] $f \\colon X \\to \\mathbb{K}$ such that $\\sup_{x \\in X} \\abs{f(x)} \u003c \\infty$.\nIt is a vector space under pointwise addition and scalar multiplication.\nIt is a [[normed space|03J5RAKB30]] with the norm\n$$\\norm{f}_\\infty = \\sup_{x \\in X} \\abs{f(x)},$$\nwith $\\norm{f}_\\infty = 0$ when $X$ is empty.\n\n## Remarks\n\nExample: $f(x) = \\sin x$ on $X = \\R$ is continuous, and $\\abs{\\sin x} \\le 1$ with equality at $x = \\pi/2$, so $f \\in C_b(\\R)$ with $\\norm{f}_\\infty = 1$.\n\nNon-example: $g(x) = x$ on $\\R$ is continuous, but $\\sup_x \\abs{g(x)} = \\infty$, so $g \\notin C_b(\\R)$.\n\nThe [[supremum|0BTMQ04556]] is finite by the definition, and the norm axioms follow from those of $\\abs{\\cdot}$.\nThe space $C_b(X)$ is complete, so it is a Banach space.\nIf $X$ is compact, $C_b(X)$ is the space $C(X)$ of all continuous functions on $X$.\nIf $X$ is a set with the discrete topology, $C_b(X)$ is the space $\\ell^\\infty(X)$ of bounded functions.\n",
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