{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "1GJ6SRT6B7"
    },
    {
      "field": "uses",
      "target": "2BDH4RQZQ9"
    },
    {
      "field": "uses",
      "target": "2CKZETC3QA"
    },
    {
      "field": "uses",
      "target": "4E4RXKV9W3"
    },
    {
      "field": "uses",
      "target": "4XQ8TYS2C4"
    },
    {
      "field": "uses",
      "target": "5J603A6DFH"
    },
    {
      "field": "uses",
      "target": "61Z8MZ2M28"
    },
    {
      "field": "uses",
      "target": "7PY20WMZDZ"
    },
    {
      "field": "uses",
      "target": "972X7QM34D"
    },
    {
      "field": "uses",
      "target": "BHMMRGCFVX"
    },
    {
      "field": "uses",
      "target": "BW21VJW403"
    },
    {
      "field": "uses",
      "target": "CE0V9686M8"
    },
    {
      "field": "uses",
      "target": "CFDZJ55N95"
    },
    {
      "field": "uses",
      "target": "CMZS6KA033"
    },
    {
      "field": "uses",
      "target": "CTYQQWJPMB"
    },
    {
      "field": "uses",
      "target": "DH3PYQJXH0"
    },
    {
      "field": "uses",
      "target": "DHYS1W3Y4N"
    },
    {
      "field": "uses",
      "target": "DS2AS9TSZP"
    },
    {
      "field": "uses",
      "target": "EHGRPCTM3J"
    },
    {
      "field": "uses",
      "target": "GPQ5M0SA47"
    },
    {
      "field": "uses",
      "target": "KXC9KHWVGE"
    },
    {
      "field": "uses",
      "target": "PFT115EGPE"
    },
    {
      "field": "uses",
      "target": "XJF5V9WHKZ"
    },
    {
      "field": "uses",
      "target": "XPCTJTBSQ0"
    },
    {
      "field": "defines",
      "target": "20YERDAS5W"
    },
    {
      "field": "about",
      "target": "CFDZJ55N95"
    },
    {
      "field": "about",
      "target": "CMZS6KA033"
    },
    {
      "field": "about",
      "target": "DS2AS9TSZP"
    },
    {
      "field": "about",
      "target": "XJF5V9WHKZ"
    },
    {
      "field": "specializes",
      "target": "CXG6JZGMCK"
    },
    {
      "field": "specializes",
      "target": "EVHCE5VR6Y"
    },
    {
      "field": "specializes",
      "target": "ZR09S9R3AW"
    }
  ],
  "body": "\n## Summary\n\nA locally compact group is a Hausdorff [[topological group|92700QAB5Y]] whose topology is [[locally compact|0FE2SB8WVJ]].\nThese groups carry a [[Haar measure|VY0HXD3GC4]], an invariant way to measure and integrate, which makes harmonic analysis on them possible.\nThey include all Lie groups, all discrete groups, all compact groups and the groups of points of algebraic groups over local fields such as $\\Q_p$.\n\n## Explanations\n\n### L1 human\n\nA space is locally compact if every point has a small compact patch around it, like a closed and bounded interval around a point of the line.\nThe real line, the circle and the $p$-adic numbers all have this property, while an infinite-dimensional Hilbert space does not.\nA locally compact group is a group whose space of elements is of this kind and whose operations are continuous.\nFor example, the group of invertible $n \\times n$ real matrices is locally compact, since it is an open subset of $\\R^{n^2}$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "22D05",
      "22A05"
    ],
    "name": "Locally compact group",
    "same_as": {
      "wikidata": "Q2147620",
      "wikipedia": "Locally compact group"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Locally_compact_group"
      }
    ],
    "specializes": [
      "7GKCGQ7215"
    ],
    "tag": "AHV05ZBCYW"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/AHV05ZBCYW/",
  "links": [
    {
      "field": "specializes",
      "target": "7GKCGQ7215"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/AHV05ZBCYW.md",
  "msc": [
    "22D05",
    "22A05"
  ],
  "name": "Locally compact group",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/AHV/AHV05ZBCYW.md",
  "status": null,
  "tag": "AHV05ZBCYW",
  "uri": "m2://AHV05ZBCYW",
  "url": "https://tamnd.github.io/math2/wiki/locally-compact-group/",
  "writing": {
    "tag": "AHV05ZBCYW",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}