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  "body": "\n## Summary\n\nFor a topological space $X$, the space $C_b(X)$ consists of the bounded continuous scalar functions on $X$, with the supremum norm.\nIt is a [[Banach space|06GZ956657]]: see [[completeness of the space of bounded continuous functions|09GRCZVR5E]] and the example of the [[sup norm|0PARCKE8WP]].\nConvergence in this norm is uniform convergence on $X$.\nWhen $X$ is compact, every continuous function is bounded and $C_b(X)$ is the classical space $C(X)$.\n\n## Explanations\n\n### L1 human\n\nThink of all continuous functions on $X$ whose graphs stay inside some horizontal band.\nThe distance between two of them is the largest vertical gap between their graphs.\nFor $X = \\R$, the functions $\\sin$ and $\\arctan$ belong to $C_b(\\R)$, but $x \\mapsto x$ does not.\nA limit of such functions, uniform over all of $X$, is again bounded and continuous.\n",
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