{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "60MQBQRACM"
    },
    {
      "field": "uses",
      "target": "D3G13ZQ2AQ"
    },
    {
      "field": "uses",
      "target": "N8SFMS441V"
    },
    {
      "field": "uses",
      "target": "Z8CW6Z2K7N"
    }
  ],
  "body": "\n## Summary\n\nA closable operator is an unbounded operator that has a closed extension.\nMany operators are first defined on a convenient small domain, such as smooth functions, and are not closed there.\nIf they are closable, they have a smallest closed extension, the closure, which is the operator one actually studies.\n\n## Explanations\n\n### L1 human\n\nSometimes a formula is written down only for nice inputs, and the resulting operator is not closed.\nOne then tries to enlarge the domain by taking limits, keeping the rule that limits of inputs go with limits of outputs.\nThis works when the same input limit can never come with two different output limits.\nFor example, differentiation on polynomials, in the space $C[0, 1]$ with the supremum norm, is closable, and its closure is differentiation on continuously differentiable functions.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "kind": "concept",
    "msc": [
      "47A05",
      "47B25"
    ],
    "name": "Closable operator",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Unbounded_operator"
      }
    ],
    "specializes": [
      "KRJX5RA0PW"
    ],
    "tag": "3TWMBFQJJ8"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/3TWMBFQJJ8/",
  "links": [
    {
      "field": "specializes",
      "target": "KRJX5RA0PW"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/3TWMBFQJJ8.md",
  "msc": [
    "47A05",
    "47B25"
  ],
  "name": "Closable operator",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/3TW/3TWMBFQJJ8.md",
  "status": null,
  "tag": "3TWMBFQJJ8",
  "uri": "m2://3TWMBFQJJ8",
  "url": "https://math2.tamnd.com/wiki/closable-operator/",
  "writing": {
    "tag": "3TWMBFQJJ8",
    "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
      }
    ]
  }
}