{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "NZ8WECT409"
    }
  ],
  "body": "\n## Summary\n\nThe Friedrichs extension is a canonical [[self-adjoint|0NEGN5CRSC]] extension of a densely defined [[symmetric operator|H971WGMV6C]] that is bounded below.\nIt is built from the quadratic form of the operator, by completing the domain in the form norm.\nIt has the same lower bound as the original operator.\nIt is the standard way to turn a formal differential operator such as $-\\Delta + V$ with $V \\ge 0$ into a self-adjoint operator.\n\n## Explanations\n\n### L1 human\n\nA symmetric operator is often given only on nice functions, and this domain is too small for the operator to be self-adjoint.\nThere can be many ways to enlarge the domain, and they give different operators.\nWhen the operator is bounded below, the Friedrichs construction picks one of them in a natural way.\nThe idea is to use the energy $\\ip{Tx}{x}$ to measure functions, and to keep exactly the functions of finite energy.\nFor $-d^2/dx^2$ on smooth functions with compact support in $(0, 1)$, the Friedrichs extension is the second derivative with the Dirichlet conditions $f(0) = f(1) = 0$.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "kind": "concept",
    "msc": [
      "47B25",
      "47A20",
      "47A07"
    ],
    "name": "Friedrichs extension",
    "same_as": {
      "wikidata": "Q1464339",
      "wikipedia": "Friedrichs extension"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Friedrichs_extension"
      }
    ],
    "tag": "GTBXZXK146"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/GTBXZXK146/",
  "links": null,
  "markdown": "https://math2.tamnd.com/node/GTBXZXK146.md",
  "msc": [
    "47B25",
    "47A20",
    "47A07"
  ],
  "name": "Friedrichs extension",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/GTB/GTBXZXK146.md",
  "status": null,
  "tag": "GTBXZXK146",
  "uri": "m2://GTBXZXK146",
  "url": "https://math2.tamnd.com/wiki/friedrichs-extension/",
  "writing": {
    "tag": "GTBXZXK146",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}