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  "body": "\n## Summary\n\nThe space $L^\\infty(\\mu)$ consists of the essentially bounded measurable functions on a measure space, up to equality almost everywhere.\nIts norm is the [[essential supremum|WPC2XRMB9Y]] of the absolute value.\nIt is the case $p = \\infty$ of the [[Lp spaces|SASP14MA1Z]] and is a Banach space.\nFor a $\\sigma$-finite measure it is the dual of $L^1(\\mu)$.\n\n## Explanations\n\n### L1 human\n\nA function lies in $L^\\infty$ when it is bounded once one ignores a set of measure zero.\nThe norm is the smallest bound that holds outside such a set.\nFor example, the function on $\\R$ equal to $\\sin x$ at irrational $x$ and to $x$ at rational $x$ is unbounded, but its $L^\\infty$ norm is $1$.\nWith the counting measure on $\\N$ one gets the space $\\ell^\\infty$ of bounded sequences.\n",
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