{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "27BDDC590S"
    },
    {
      "field": "about",
      "target": "AJ5RG9XQYZ"
    },
    {
      "field": "about",
      "target": "G1VWNQSAK5"
    },
    {
      "field": "uses",
      "target": "AJ5RG9XQYZ"
    },
    {
      "field": "uses",
      "target": "DYAWGXE2GV"
    },
    {
      "field": "uses",
      "target": "G1VWNQSAK5"
    },
    {
      "field": "uses",
      "target": "NWK9Z80BT2"
    },
    {
      "field": "uses",
      "target": "S49XMZKGQC"
    }
  ],
  "body": "\n## Summary\n\nA normed space is reflexive when the [[canonical embedding into its bidual|QKR3D65TTH]] is onto.\nReflexive spaces are always [[Banach spaces|06GZ956657]].\nThey are the Banach spaces whose closed unit ball is weakly compact.\nSo bounded sequences have weakly convergent subsequences, which makes them the natural setting for existence proofs by minimization.\nHilbert spaces, $L^p$ spaces for $1 \u003c p \u003c \\infty$ and finite-dimensional spaces are reflexive.\n\n## Explanations\n\n### L1 human\n\nEvery normed space sits inside its double dual, the dual of the dual.\nIn a reflexive space the double dual has nothing more: going to the dual twice brings you back to where you started.\nFor example, the dual of $\\ell^2$ is $\\ell^2$ again, so $\\ell^2$ is reflexive.\nThe space $c_0$ of null sequences is not reflexive: its double dual is the bigger space $\\ell^\\infty$ of all bounded sequences.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "kind": "concept",
    "msc": [
      "46B10"
    ],
    "name": "Reflexive space",
    "same_as": {
      "wikidata": "Q2032198",
      "wikipedia": "Reflexive space"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Reflexive_space"
      }
    ],
    "specializes": [
      "06GZ956657"
    ],
    "tag": "NN8RWPC8WG"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/NN8RWPC8WG/",
  "links": [
    {
      "field": "specializes",
      "target": "06GZ956657"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/NN8RWPC8WG.md",
  "msc": [
    "46B10"
  ],
  "name": "Reflexive space",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/NN8/NN8RWPC8WG.md",
  "status": null,
  "tag": "NN8RWPC8WG",
  "uri": "m2://NN8RWPC8WG",
  "url": "https://math2.tamnd.com/wiki/reflexive-space/",
  "writing": {
    "tag": "NN8RWPC8WG",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}