{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $X$ be a [[separable Banach space|AH45GS6RCR]] over $\\R$.\nLet $C([0,1])$ be the Banach space of continuous real functions on $[0,1]$ with the [[supremum norm|0PARCKE8WP]].\nThen there is a [[linear isometry|07SJXTWVQW]] $T \\colon X \\to C([0,1])$.\nIn particular, $X$ is isometrically isomorphic to the closed subspace $T(X)$ of $C([0,1])$.\n\n## Remarks\n\nThe idea of the proof: let $B$ be the closed unit ball of the dual space $X^*$ with the weak-* topology.\nSince $X$ is separable, $B$ is a compact metrizable space, so it is a continuous image of the Cantor set $\\Delta$ under some map $\\varphi$.\nThe map $x \\mapsto (t \\mapsto \\varphi(t)(x))$ is a linear isometry of $X$ into $C(\\Delta)$, by the Hahn–Banach theorem.\nEach function in $C(\\Delta)$ extends to $[0,1]$ by linear interpolation on the open intervals removed from $[0,1]$, and this extension is a linear isometry from $C(\\Delta)$ into $C([0,1])$.\n\nFor example, the Hilbert space $\\ell^2$ and every space $L^p([0,1])$ with $1 \\le p \u003c \\infty$ are isometric to subspaces of $C([0,1])$.\nSeparability is necessary: $C([0,1])$ is separable, so each of its subspaces is separable, and the non-separable space $\\ell^\\infty$ does not embed in it.\n\n## Explanations\n\n### L1 human\n\nThe space of continuous functions on an interval is big enough to hold a copy of every separable Banach space.\nThe copy keeps all distances exactly.\nSo questions about separable Banach spaces can be asked inside one single space.\n\n### L2 human\n\nThe theorem says that $C([0,1])$ is universal for the class of separable Banach spaces.\nIt is the linear analogue of the fact that every compact metric space is a continuous image of the Cantor set, and the proof uses exactly that fact.\nThe weak-* compactness of the dual ball comes from the Banach–Alaoglu theorem.\nUniversality makes $C([0,1])$ itself very complicated: every property that fails in some separable space fails in some subspace of $C([0,1])$.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "about": [
      "AH45GS6RCR"
    ],
    "kind": "statement",
    "msc": [
      "46B04",
      "46B25"
    ],
    "name": "Banach–Mazur theorem",
    "role": "theorem",
    "same_as": {
      "wikidata": "Q2226610",
      "wikipedia": "Banach–Mazur theorem"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Banach%E2%80%93Mazur_theorem"
      }
    ],
    "tag": "QC5MB5WM8F",
    "uses": [
      "AH45GS6RCR",
      "07SJXTWVQW",
      "0PARCKE8WP"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/QC5MB5WM8F/",
  "links": [
    {
      "field": "about",
      "target": "AH45GS6RCR"
    },
    {
      "field": "uses",
      "target": "AH45GS6RCR"
    },
    {
      "field": "uses",
      "target": "07SJXTWVQW"
    },
    {
      "field": "uses",
      "target": "0PARCKE8WP"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/QC5MB5WM8F.md",
  "msc": [
    "46B04",
    "46B25"
  ],
  "name": "Banach–Mazur theorem",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/QC5/QC5MB5WM8F.md",
  "status": null,
  "tag": "QC5MB5WM8F",
  "uri": "m2://QC5MB5WM8F",
  "url": "https://math2.tamnd.com/wiki/banach-mazur-theorem/",
  "writing": {
    "tag": "QC5MB5WM8F",
    "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
      }
    ]
  }
}