{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "NY6FVCX61P"
    },
    {
      "field": "uses",
      "target": "ZXJ7FJ3WCE"
    }
  ],
  "body": "\n## Statement\n\nLet $X$ and $Y$ be [[Banach spaces|06GZ956657]] over $\\R$ or $\\C$, and let $T \\colon X \\to Y$ be a [[linear map|0NRYHJPQG2]].\nGive $X \\times Y$ the norm $\\norm{(x, y)} = \\norm{x} + \\norm{y}$.\nThen $T$ is [[continuous|03TM135G0N]] if and only if its [[graph|0JBJYT69ZV]] $\\Gamma(T) = \\set{(x, Tx) : x \\in X}$ is [[closed|0ZCK8DQYN1]] in $X \\times Y$.\n\n## Remarks\n\nIf $T$ is continuous, its graph is closed because $Y$ is Hausdorff.\nConversely, if $\\Gamma(T)$ is closed, it is a Banach space, and the projection $(x, Tx) \\mapsto x$ is a continuous linear bijection from $\\Gamma(T)$ onto $X$.\nThe [[bounded inverse theorem|MR5FJXR348]] makes its inverse $x \\mapsto (x, Tx)$ continuous, and so $T$ is continuous.\n\nIn practice the closed graph condition is easier to check: it says that if $x_n \\to x$ and $T x_n \\to y$, then $y = Tx$.\nOne does not have to show that $T x_n$ converges.\nCompleteness is necessary.\nLet $X = C^1[0, 1]$ and $Y = C[0, 1]$, both with the sup norm, and let $T f = f'$.\nIf $f_n \\to f$ and $f_n' \\to g$ uniformly, then $f$ is differentiable with $f' = g$, so the graph is closed.\nBut $T$ is not continuous, since $f_n(t) = t^n$ has $\\norm{f_n} = 1$ and $\\norm{f_n'} = n$; here $X$ is not complete.\n\n## Explanations\n\n### L1 human\n\nTo show that a linear map between complete normed spaces is continuous, it is enough to show that its graph is a closed set.\nThis means: whenever inputs converge and outputs converge, the limit output is the image of the limit input.\nThat is often much easier than finding a bound directly.\n\n### L2 human\n\nThe theorem is used to prove automatic continuity: for example, a linear map $T$ defined on a whole Hilbert space with $\\ip{Tx}{y} = \\ip{x}{Ty}$ for all $x, y$ is bounded (Hellinger–Toeplitz theorem).\nIt is also used to show that a projection onto a closed subspace along a closed complement is bounded, see [[complemented subspace|Q0TY362K0V]].\nUnbounded operators in quantum mechanics are typically closed operators defined only on dense subspaces, which is consistent with the theorem.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "about": [
      "03TM135G0N",
      "0JBJYT69ZV"
    ],
    "kind": "statement",
    "msc": [
      "46A30",
      "47A05"
    ],
    "name": "Closed graph theorem",
    "role": "theorem",
    "same_as": {
      "wikidata": "Q104845546",
      "wikipedia": "Closed graph theorem (functional analysis)"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Closed_graph_theorem_(functional_analysis)"
      }
    ],
    "tag": "GXM8BX8YM7",
    "uses": [
      "06GZ956657",
      "0NRYHJPQG2",
      "03TM135G0N",
      "0JBJYT69ZV",
      "0ZCK8DQYN1"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/GXM8BX8YM7/",
  "links": [
    {
      "field": "about",
      "target": "03TM135G0N"
    },
    {
      "field": "about",
      "target": "0JBJYT69ZV"
    },
    {
      "field": "uses",
      "target": "06GZ956657"
    },
    {
      "field": "uses",
      "target": "0NRYHJPQG2"
    },
    {
      "field": "uses",
      "target": "03TM135G0N"
    },
    {
      "field": "uses",
      "target": "0JBJYT69ZV"
    },
    {
      "field": "uses",
      "target": "0ZCK8DQYN1"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/GXM8BX8YM7.md",
  "msc": [
    "46A30",
    "47A05"
  ],
  "name": "Closed graph theorem",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/GXM/GXM8BX8YM7.md",
  "status": null,
  "tag": "GXM8BX8YM7",
  "uri": "m2://GXM8BX8YM7",
  "url": "https://math2.tamnd.com/wiki/closed-graph-theorem/",
  "writing": {
    "tag": "GXM8BX8YM7",
    "kind": "statement",
    "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
      }
    ]
  }
}