{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $(X, \\Sigma, \\mu)$ be a measure space, let $1 \\le p \u003c \\infty$, and let $q \\in (1, \\infty]$ be the conjugate exponent, with $1/p + 1/q = 1$.\nIf $p = 1$, assume moreover that $\\mu$ is [[σ-finite|WEBBT18D4X]].\nFor $g \\in L^q(\\mu)$, let $\\Lambda_g$ be the functional on [[$L^p(\\mu)$|SASP14MA1Z]] given by\n$$\\Lambda_g(f) = \\int_X f g \\, d\\mu.$$\nThen $g \\mapsto \\Lambda_g$ is an [[isometric isomorphism|0ZXSF2YQV2]] of $L^q(\\mu)$ onto the [[dual space|07JBSZE0V4]] $L^p(\\mu)^*$.\n\n## Remarks\n\nHölder's inequality gives $\\abs{\\Lambda_g(f)} \\le \\norm{f}_p \\norm{g}_q$, so $\\norm{\\Lambda_g} \\le \\norm{g}_q$.\nFor $1 \u003c p \u003c \\infty$, the choice $f = \\abs{g}^{q-1} \\overline{\\sgn g}$ gives equality, so the map is an isometry.\nSurjectivity is the main point.\nLet $\\mu$ be $\\sigma$-finite and let $\\Lambda$ be a bounded functional.\nThen $E \\mapsto \\Lambda(\\mathbf{1}_E)$, on sets of finite measure, is a set function absolutely continuous with respect to $\\mu$.\nThe [[Radon–Nikodym theorem|1ZSZS5G1Q4]] gives $g$.\nFor $1 \u003c p \u003c \\infty$ and a general measure space, one reduces to the $\\sigma$-finite case, or uses the uniform convexity of $L^p$.\n\nExample: for counting measure on $\\N$, the theorem says that $(\\ell^p)^* = \\ell^q$ for $1 \\le p \u003c \\infty$.\n\nThe $\\sigma$-finiteness for $p = 1$ cannot be dropped.\nLet $X = \\set{x}$ with $\\mu(\\set{x}) = \\infty$.\nThen $L^1(\\mu) = \\set{0}$, whose dual is $\\set{0}$, but $L^\\infty(\\mu)$ is the one-dimensional space of constants.\n\nThe case $p = \\infty$ is excluded.\nThe map $g \\mapsto \\Lambda_g$ from $L^1$ into $(L^\\infty)^*$ is an isometry, but in general it is not onto.\nFor example, take the functional on convergent sequences that gives the limit.\nBy the Hahn–Banach theorem it extends to a bounded functional on $\\ell^\\infty$, and this extension is not given by any element of $\\ell^1$.\n\n## Explanations\n\n### L1 human\n\nEvery continuous linear way of turning a function in $L^p$ into a number is integration against a fixed function $g$.\nThe function $g$ lives in the partner space $L^q$, where $1/p + 1/q = 1$.\nSo $L^2$ is its own partner, $L^1$ is paired with $L^\\infty$, and $L^{3/2}$ is paired with $L^3$.\n\n### L2 human\n\nThe theorem identifies $(L^p)^* = L^q$ for $1 \\le p \u003c \\infty$ under the stated hypotheses.\nFor $1 \u003c p \u003c \\infty$ it follows that $L^p$ is reflexive, since $(L^p)^{**} = (L^q)^* = L^p$.\nWeak convergence in $L^p$ then means convergence of $\\int f_n g \\, d\\mu$ for each $g \\in L^q$, and bounded sequences have weakly convergent subsequences.\nNeither $L^1$ nor $L^\\infty$ is reflexive in general, which is the source of many technical difficulties.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "about": [
      "SASP14MA1Z",
      "07JBSZE0V4"
    ],
    "kind": "statement",
    "msc": [
      "46E30",
      "46B10"
    ],
    "name": "Dual of Lp",
    "role": "theorem",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Lp_space"
      }
    ],
    "tag": "Q1V7ZNS332",
    "uses": [
      "SASP14MA1Z",
      "07JBSZE0V4",
      "0ZXSF2YQV2",
      "WEBBT18D4X"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/Q1V7ZNS332/",
  "links": [
    {
      "field": "about",
      "target": "SASP14MA1Z"
    },
    {
      "field": "about",
      "target": "07JBSZE0V4"
    },
    {
      "field": "uses",
      "target": "SASP14MA1Z"
    },
    {
      "field": "uses",
      "target": "07JBSZE0V4"
    },
    {
      "field": "uses",
      "target": "0ZXSF2YQV2"
    },
    {
      "field": "uses",
      "target": "WEBBT18D4X"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/Q1V7ZNS332.md",
  "msc": [
    "46E30",
    "46B10"
  ],
  "name": "Dual of Lp",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/Q1V/Q1V7ZNS332.md",
  "status": null,
  "tag": "Q1V7ZNS332",
  "uri": "m2://Q1V7ZNS332",
  "url": "https://math2.tamnd.com/wiki/dual-of-lp/",
  "writing": {
    "tag": "Q1V7ZNS332",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}