{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "5D5SC7NNJ2"
    }
  ],
  "body": "\n## Summary\n\nThe Pettis integral of a function with values in a Banach space is a vector in that space.\nEach continuous linear functional maps it to the integral of the corresponding scalar function.\nIt is a weak notion of integral, defined through the dual space.\nEvery Bochner integrable function is Pettis integrable with the same integral, but in infinite dimensions there are Pettis integrable functions that are not Bochner integrable.\nIt is also called the Gelfand–Pettis integral.\n\n## Explanations\n\n### L1 human\n\nTo integrate a vector-valued function, one can integrate each coordinate.\nIn an infinite-dimensional space, the role of coordinates is played by continuous linear functionals.\nThe Pettis integral is a vector that has the right value in every such coordinate, for every measurable set.\nFor a function into $\\R^2$, it is the vector whose two coordinates are the integrals of the two components of the function.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "kind": "concept",
    "msc": [
      "28B05",
      "46G10"
    ],
    "name": "Pettis integral",
    "same_as": {
      "wikidata": "Q211188",
      "wikipedia": "Pettis integral"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Pettis_integral"
      }
    ],
    "tag": "9A5F3JGTC0"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/9A5F3JGTC0/",
  "links": null,
  "markdown": "https://math2.tamnd.com/node/9A5F3JGTC0.md",
  "msc": [
    "28B05",
    "46G10"
  ],
  "name": "Pettis integral",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/9A5/9A5F3JGTC0.md",
  "status": null,
  "tag": "9A5F3JGTC0",
  "uri": "m2://9A5F3JGTC0",
  "url": "https://math2.tamnd.com/wiki/pettis-integral/",
  "writing": {
    "tag": "9A5F3JGTC0",
    "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
      }
    ]
  }
}