{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "VWFVT9AGCT"
    }
  ],
  "body": "\n## Summary\n\nA convenient vector space is a [[locally convex space|B6GY2KJMWW]] that is complete in a weak sense: every Mackey–Cauchy sequence converges.\nThese spaces are the setting of the convenient calculus of Frölicher, Kriegl and Michor.\nIn it, a map is smooth when it sends smooth curves to smooth curves.\nThe category of convenient spaces and smooth maps is cartesian closed.\n\nThis makes spaces of smooth maps, and manifolds of mappings, easy to handle without Banach-space estimates.\n\n## Explanations\n\n### L1 human\n\nCalculus beyond Banach spaces has a problem: there are many inequivalent notions of a derivative.\nThe convenient approach tests smoothness only along curves: a map is smooth if every smooth path is sent to a smooth path.\nFor this to work, the space must contain enough limits, at least those that integrating a smooth curve produces.\nConvenient vector spaces are the spaces with exactly this much completeness.\nEvery Banach space and every [[Fréchet space|3GAA614673]] is convenient.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "46A17",
      "58C20",
      "46T20"
    ],
    "name": "Convenient vector space",
    "same_as": {
      "wikidata": "Q25303670",
      "wikipedia": "Convenient vector space"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Convenient_vector_space"
      }
    ],
    "specializes": [
      "B6GY2KJMWW"
    ],
    "tag": "46MSX1FXZT"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/46MSX1FXZT/",
  "links": [
    {
      "field": "specializes",
      "target": "B6GY2KJMWW"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/46MSX1FXZT.md",
  "msc": [
    "46A17",
    "58C20",
    "46T20"
  ],
  "name": "Convenient vector space",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/46M/46MSX1FXZT.md",
  "status": null,
  "tag": "46MSX1FXZT",
  "uri": "m2://46MSX1FXZT",
  "url": "https://tamnd.github.io/math2/wiki/convenient-vector-space/",
  "writing": {
    "tag": "46MSX1FXZT",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}