{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $X$ be a real or complex [[Banach space|06GZ956657]] with [[continuous dual|4A82S5FVRT]] $X^*$.\nThe space $X$ has the *Dunford–Pettis property* if the following holds.\nLet $(x_n)$ be a sequence in $X$ that converges to $0$ in the [[weak topology|P6KHVSTPZV]] $\\sigma(X, X^*)$.\nLet $(f_n)$ be a sequence in $X^*$ that converges to $0$ in the weak topology $\\sigma(X^*, X^{**})$.\nThen $f_n(x_n) \\to 0$.\n\n## Remarks\n\nExample: $\\ell^1$ has the Dunford–Pettis property.\nBy the Schur property of $\\ell^1$, a weakly null sequence $(x_n)$ in $\\ell^1$ tends to $0$ in norm.\nA weakly convergent sequence $(f_n)$ is bounded, by the uniform boundedness principle.\nSo $\\abs{f_n(x_n)} \\le \\norm{f_n} \\, \\norm{x_n} \\to 0$.\n\nNon-example: $\\ell^2$ does not have it.\nThe unit vectors $e_n$ tend weakly to $0$, and so do the functionals $f_n = \\ip{\\cdot}{e_n}$ in the dual of $\\ell^2$.\nBut $f_n(e_n) = 1$ for all $n$.\nMore generally, no infinite-dimensional reflexive space has the property.\n\nAn equivalent form: every weakly compact operator from $X$ to a Banach space $Y$ maps weakly convergent sequences to norm convergent sequences.\nThe spaces $L^1(\\mu)$, $C(K)$ for compact Hausdorff $K$, and $c_0$ have the property.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "defines": "QQSNZX1QMM",
    "kind": "definition",
    "msc": [
      "46B10",
      "46B20"
    ],
    "name": "Dunford–Pettis property",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Dunford%E2%80%93Pettis_property"
      }
    ],
    "tag": "W35ZP39S4R",
    "uses": [
      "06GZ956657",
      "4A82S5FVRT",
      "P6KHVSTPZV"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/W35ZP39S4R/",
  "links": [
    {
      "field": "defines",
      "target": "QQSNZX1QMM"
    },
    {
      "field": "uses",
      "target": "06GZ956657"
    },
    {
      "field": "uses",
      "target": "4A82S5FVRT"
    },
    {
      "field": "uses",
      "target": "P6KHVSTPZV"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/W35ZP39S4R.md",
  "msc": [
    "46B10",
    "46B20"
  ],
  "name": "Dunford–Pettis property",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/W35/W35ZP39S4R.md",
  "status": null,
  "tag": "W35ZP39S4R",
  "uri": "m2://W35ZP39S4R",
  "url": "https://math2.tamnd.com/wiki/dunford-pettis-property-definition/",
  "writing": {
    "tag": "W35ZP39S4R",
    "kind": "definition",
    "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
      }
    ]
  }
}