{
  "aliases": [
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  "body": "\n## Example\n\nLet $X$ be a [[topological space|0CNQXVBHCX]] and let $C_b(X)$ be the real vector space of bounded [[continuous maps|0SY9ECQ19Q]] $f \\colon X \\to \\mathbb{R}$.\nFor $f \\in C_b(X)$, let $\\|f\\|_\\infty = \\sup_{x \\in X} |f(x)|$, with $\\|f\\|_\\infty = 0$ when $X$ is empty.\nThen $\\|\\cdot\\|_\\infty$ is a norm on $C_b(X)$, the sup norm, and $C_b(X)$ is a [[Banach space|0BDTTM24DP]] with this norm.\n\n## Remarks\n\nThe [[supremum|0BTMQ04556]] is finite because $f$ is bounded.\nThe metric $\\|f - g\\|_\\infty$ is the [[uniform metric|06HE9J7DXF]]. Its completeness is [[completeness of the space of bounded continuous functions|09GRCZVR5E]].\nA sequence converges in the sup norm if and only if it converges uniformly.\nFor a compact space $X$, each continuous real function is bounded, so $C_b(X) = C(X)$.\n\n## Explanations\n\n### L1 human\n\nThe sup norm of a function is the height of the highest point of its graph, in absolute value.\nTwo functions are close in this norm when their graphs are close everywhere.\n",
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