{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Counterexample\n\nThere is a separable [[Banach space|06GZ956657]] $X$ that does not have the [[approximation property|8CKE0V6ANK]].\nThat is, there are a compact set $K \\subseteq X$ and $\\varepsilon \u003e 0$ with the following property.\nEvery continuous linear map $T \\colon X \\to X$ of finite rank satisfies $\\norm{Tx - x} \\ge \\varepsilon$ for some $x \\in K$.\nPer Enflo constructed such a space; his paper appeared in Acta Mathematica in 1973.\n\n## Remarks\n\nThe construction is an explicit but intricate combinatorial one, built from finite-dimensional blocks; it is not short enough to reproduce here.\nIt has two famous consequences.\nFirst, $X$ has no Schauder basis, because a space with a Schauder basis has the approximation property.\nSo the basis problem, whether every separable Banach space has a Schauder basis, has a negative answer.\nSecond, by Grothendieck's work on the approximation property, some compact operator into $X$ is not a norm limit of finite-rank operators.\nLater work showed that the spaces $\\ell^p$ with $p \\ne 2$ contain closed subspaces without the approximation property.\n\n## Explanations\n\n### L1 human\n\nFor a long time it was hoped that every reasonable Banach space could be described by coordinates, like the sequence spaces.\nIt was also hoped that every compact operator was a limit of matrices.\nEnflo's space shows that both hopes are false.\nIt is a separable Banach space where the identity cannot be imitated on compact sets by finite-rank maps.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "kind": "counterexample",
    "msc": [
      "46B28",
      "46B15"
    ],
    "name": "Enflo's counterexample to the approximation problem",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Per_Enflo"
      },
      {
        "url": "https://en.wikipedia.org/wiki/Approximation_property"
      }
    ],
    "tag": "ZKP5KGFWCK",
    "uses": [
      "06GZ956657",
      "8CKE0V6ANK"
    ]
  },
  "kind": "counterexample",
  "link": "https://math2.tamnd.com/node/ZKP5KGFWCK/",
  "links": [
    {
      "field": "uses",
      "target": "06GZ956657"
    },
    {
      "field": "uses",
      "target": "8CKE0V6ANK"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/ZKP5KGFWCK.md",
  "msc": [
    "46B28",
    "46B15"
  ],
  "name": "Enflo's counterexample to the approximation problem",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/ZKP/ZKP5KGFWCK.md",
  "status": null,
  "tag": "ZKP5KGFWCK",
  "uri": "m2://ZKP5KGFWCK",
  "url": "https://math2.tamnd.com/wiki/enflos-counterexample-to-the-approximation-problem/",
  "writing": {
    "tag": "ZKP5KGFWCK",
    "kind": "counterexample",
    "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
      },
      {
        "what": "edge refutes or not_instance_of out",
        "have": 0,
        "want": 1
      }
    ]
  }
}