{
  "aliases": null,
  "backlinks": [
    {
      "field": "instance_of",
      "target": "CFHNV77ZMD"
    },
    {
      "field": "about",
      "target": "JWCDDMJE3K"
    },
    {
      "field": "uses",
      "target": "JWCDDMJE3K"
    },
    {
      "field": "uses",
      "target": "M2W79VF0TN"
    },
    {
      "field": "uses",
      "target": "Q9VG3EGT74"
    },
    {
      "field": "defines",
      "target": "P4X779XNEA"
    }
  ],
  "body": "\n## Summary\n\nA Hilbert manifold is a [[Banach manifold|9K15AMGBV9]] whose charts take values in [[Hilbert spaces|HM2HHE3J51]].\nInner products give partitions of unity, orthogonal complements and Riemannian metrics, so much of finite-dimensional differential topology carries over.\nSpaces of paths and loops of Sobolev class $H^1$ in a finite-dimensional manifold are Hilbert manifolds.\nThis makes them the natural setting for Morse theory in infinite dimensions.\nTake an infinite-dimensional separable model and a separable metrizable manifold.\nThen the [[open embedding theorem|JWCDDMJE3K]] shows that it is an open subset of its model space, up to diffeomorphism.\n\n## Explanations\n\n### L1 human\n\nA Hilbert manifold is built from patches that look like pieces of a Hilbert space, glued by smooth maps.\nA Hilbert space is a possibly infinite-dimensional space in which lengths and angles make sense, such as the space $\\ell^2$ of square-summable sequences.\nThe unit sphere of $\\ell^2$, the set of sequences with $\\sum x_n^2 = 1$, is a Hilbert manifold.\nNear each of its points it looks like the hyperplane orthogonal to that point.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "58B99",
      "46T05",
      "57N20"
    ],
    "name": "Hilbert manifold",
    "same_as": {
      "wikidata": "Q3554824",
      "wikipedia": "Hilbert manifold"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Hilbert_manifold"
      }
    ],
    "specializes": [
      "9K15AMGBV9"
    ],
    "tag": "R8DMMW0E9M"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/R8DMMW0E9M/",
  "links": [
    {
      "field": "specializes",
      "target": "9K15AMGBV9"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/R8DMMW0E9M.md",
  "msc": [
    "58B99",
    "46T05",
    "57N20"
  ],
  "name": "Hilbert manifold",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/R8D/R8DMMW0E9M.md",
  "status": null,
  "tag": "R8DMMW0E9M",
  "uri": "m2://R8DMMW0E9M",
  "url": "https://tamnd.github.io/math2/wiki/hilbert-manifold/",
  "writing": {
    "tag": "R8DMMW0E9M",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}