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  "body": "\n## Statement\n\nLet $(\\Omega, \\Sigma, \\mu)$ be a [[measure space|P15VQS5DJ9]] and let $1 \\le p \u003c \\infty$.\nLet $L^p(\\mu)$ be the [[Lp space|SASP14MA1Z]] of real or complex functions.\nLet $q$ be the conjugate exponent, with $1/p + 1/q = 1$ and $q = \\infty$ when $p = 1$.\nFor $g \\in L^q(\\mu)$, let $\\Phi(g)$ be the functional $f \\mapsto \\int_\\Omega f g \\, d\\mu$ on $L^p(\\mu)$.\n\n1. If $1 \u003c p \u003c \\infty$, then $\\Phi$ is a [[linear isometry|07SJXTWVQW]] of $L^q(\\mu)$ onto the [[continuous dual|4A82S5FVRT]] of $L^p(\\mu)$.\n2. If $p = 1$ and $\\mu$ is [[σ-finite|WEBBT18D4X]], then $\\Phi$ is a linear isometry of $L^\\infty(\\mu)$ onto the continuous dual of $L^1(\\mu)$.\n\n## Remarks\n\nBy [[Hölder's inequality|JQ9WXC729D]], $\\abs{\\Phi(g)(f)} \\le \\norm{f}_p \\norm{g}_q$, so $\\Phi(g)$ is bounded with norm at most $\\norm{g}_q$.\nThe test function $f = \\abs{g}^{q-1} \\overline{\\sgn g}$ gives equality when $1 \u003c p \u003c \\infty$.\nSurjectivity is the hard part: for σ-finite $\\mu$ it follows from the Radon–Nikodym theorem, and for $1 \u003c p \u003c \\infty$ uniform convexity of $L^p$ removes the σ-finiteness.\n\nFor counting measure on $\\N$, the theorem says that the dual of $\\ell^p$ is $\\ell^q$, and the dual of $\\ell^1$ is $\\ell^\\infty$.\nThe case $p = \\infty$ is excluded: the dual of $\\ell^\\infty$ is strictly larger than $\\ell^1$.\nFor example, a Banach limit is a bounded functional on $\\ell^\\infty$ that is not given by any sequence in $\\ell^1$.\nFor $p = 1$, the σ-finite hypothesis cannot be dropped in general.\n\n## Explanations\n\n### L1 human\n\nEvery continuous linear way to turn an $L^p$ function into a number is integration against a fixed function from $L^q$.\nThat function is unique, and its size is exactly the size of the functional.\nThis turns abstract functionals into concrete functions.\n\n### L2 human\n\nThe theorem shows that $L^p$ is reflexive for $1 \u003c p \u003c \\infty$, since the dual of $L^q$ is again $L^p$.\nIn the case $p = 2$ it is a special case of the Riesz representation theorem for Hilbert spaces.\nIt is used to identify weak limits, to define weak solutions of equations and to prove inequalities by duality, where one computes $\\norm{f}_p$ as a supremum of $\\abs{\\int f g}$ over $\\norm{g}_q \\le 1$.\n",
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