{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $X$ and $Y$ be [[compact|0TVJZYB6SH]] [[Hausdorff spaces|0S1FJ8VEX9]].\nLet $C(X)$ and $C(Y)$ be the spaces of continuous real functions with the [[supremum norm|0PARCKE8WP]].\nLet $T \\colon C(X) \\to C(Y)$ be a surjective [[linear isometry|07SJXTWVQW]].\nThen there are a [[homeomorphism|0QT00PB5G2]] $\\varphi \\colon Y \\to X$ and a continuous function $h \\colon Y \\to \\set{-1, 1}$ such that\n$$(T f)(y) = h(y) \\, f(\\varphi(y)) \\quad \\text{for all } f \\in C(X), \\ y \\in Y.$$\n\n## Remarks\n\nThe idea of the proof: the extreme points of the unit ball of the dual of $C(X)$ are the functionals $\\pm \\delta_x$, with $\\delta_x(f) = f(x)$.\nThe adjoint of $T$ is an isometry of the duals and maps extreme points to extreme points.\nSo it sends each $\\delta_y$ to $h(y) \\delta_{\\varphi(y)}$, and the weak-* topology makes $\\varphi$ and $h$ continuous.\n\nExample: on $X = Y = [0, 1]$, the map $(T f)(t) = -f(1 - t)$ is a surjective linear isometry.\nHere $h = -1$ and $\\varphi(t) = 1 - t$.\n\nFor complex scalars the same holds with $h$ a continuous function of absolute value $1$.\nAs a consequence, $X$ and $Y$ are homeomorphic if and only if $C(X)$ and $C(Y)$ are isometrically isomorphic.\nIsometry is needed.\nLet $S = \\set{0} \\cup \\set{1/n : n \\ge 1}$, a convergent sequence with its limit.\nThen $C(S)$ and $C(S \\times \\set{1, 2})$ are linearly isomorphic, but $S$ has one limit point and $S \\times \\set{1, 2}$ has two.\n\n## Explanations\n\n### L1 human\n\nThe norm of $C(X)$ remembers the space $X$.\nEvery way to match $C(X)$ with $C(Y)$ that keeps distances comes from matching the points of $Y$ with the points of $X$, up to a change of sign.\n\n### L2 human\n\nThe theorem shows that the Banach space $C(X)$, with its norm, determines the compact space $X$ up to homeomorphism.\nGelfand–Naimark theory gives a similar result for $C(X)$ as an algebra, and the Banach–Stone theorem gets it from the metric structure alone.\nWithout the isometry condition the result fails badly: by a theorem of Milutin, $C(X)$ is isomorphic to $C([0, 1])$ for every uncountable compact metric space $X$.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "about": [
      "MG3X8Y1VT4"
    ],
    "kind": "statement",
    "msc": [
      "46B04",
      "46E15",
      "46J10"
    ],
    "name": "Banach–Stone theorem",
    "role": "theorem",
    "same_as": {
      "wikidata": "Q4853767",
      "wikipedia": "Banach–Stone theorem"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Banach%E2%80%93Stone_theorem"
      }
    ],
    "tag": "Q8FERXTZ8B",
    "uses": [
      "0TVJZYB6SH",
      "0S1FJ8VEX9",
      "0PARCKE8WP",
      "07SJXTWVQW",
      "0QT00PB5G2"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/Q8FERXTZ8B/",
  "links": [
    {
      "field": "about",
      "target": "MG3X8Y1VT4"
    },
    {
      "field": "uses",
      "target": "0TVJZYB6SH"
    },
    {
      "field": "uses",
      "target": "0S1FJ8VEX9"
    },
    {
      "field": "uses",
      "target": "0PARCKE8WP"
    },
    {
      "field": "uses",
      "target": "07SJXTWVQW"
    },
    {
      "field": "uses",
      "target": "0QT00PB5G2"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/Q8FERXTZ8B.md",
  "msc": [
    "46B04",
    "46E15",
    "46J10"
  ],
  "name": "Banach–Stone theorem",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/Q8F/Q8FERXTZ8B.md",
  "status": null,
  "tag": "Q8FERXTZ8B",
  "uri": "m2://Q8FERXTZ8B",
  "url": "https://math2.tamnd.com/wiki/banach-stone-theorem/",
  "writing": {
    "tag": "Q8FERXTZ8B",
    "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
      }
    ]
  }
}