{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "W35ZP39S4R"
    }
  ],
  "body": "\n## Summary\n\nA [[Banach space|06GZ956657]] has the Dunford–Pettis property when the following holds.\nPairing a weakly null sequence of vectors with a weakly null sequence of functionals always gives a null sequence of numbers.\nThe spaces $L^1(\\mu)$ and $C(K)$ have it, by results of Dunford and Pettis and of Grothendieck.\nNo infinite-dimensional [[reflexive space|NN8RWPC8WG]] has it.\nThe property is a tool to show that some operators on these classical spaces are compact.\n\n## Explanations\n\n### L1 human\n\nIn infinite dimensions a sequence can tend to zero in a weak sense while its points stay far from zero.\nThe same happens to sequences of functionals.\nIn many spaces, two such sequences can still meet: the values $f_n(x_n)$ need not go to zero.\nIn $\\ell^2$, the unit vectors $e_n$ and the functionals \"take the $n$-th coordinate\" are both weakly null, yet each value is $1$.\nSpaces with the Dunford–Pettis property forbid this.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "kind": "concept",
    "msc": [
      "46B10",
      "46B20"
    ],
    "name": "Dunford–Pettis property",
    "same_as": {
      "wikidata": "Q1265637",
      "wikipedia": "Dunford–Pettis property"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Dunford%E2%80%93Pettis_property"
      }
    ],
    "tag": "QQSNZX1QMM"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/QQSNZX1QMM/",
  "links": null,
  "markdown": "https://math2.tamnd.com/node/QQSNZX1QMM.md",
  "msc": [
    "46B10",
    "46B20"
  ],
  "name": "Dunford–Pettis property",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/QQS/QQSNZX1QMM.md",
  "status": null,
  "tag": "QQSNZX1QMM",
  "uri": "m2://QQSNZX1QMM",
  "url": "https://math2.tamnd.com/wiki/dunford-pettis-property/",
  "writing": {
    "tag": "QQSNZX1QMM",
    "kind": "concept",
    "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
      }
    ]
  }
}