"locally compact group" meaning in All languages combined

{
  "forms": [
    {
      "form": "locally compact groups",
      "tags": [
        "plural"
      ]
    }
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  "head_templates": [
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      "expansion": "locally compact group (plural locally compact groups)",
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  ],
  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
  "senses": [
    {
      "categories": [
        {
          "kind": "other",
          "name": "English entries with incorrect language header",
          "parents": [
            "Entries with incorrect language header",
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          "source": "w"
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          "kind": "topical",
          "langcode": "en",
          "name": "Topology",
          "orig": "en:Topology",
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      "examples": [
        {
          "ref": "1988, J. M. G. Fell, R. S. Doran, Representations of *-Algebras, Locally Compact Groups, and Banach *-Algebraic Bundles, Volume 1, Academic Press, page 63",
          "text": "Indeed, if there is one property of locally compact groups more responsible than any other for the rich positive content of their representation theory, it is their possession of left-invariant and right-invariant (Haar) measures. The connection between the representation theory of a locally compact group and that of general Banach algebras proceeds directly from Lebesgue integration with respect to Haar measure on the group.",
          "type": "quotation"
        },
        {
          "ref": "2012, H. Heyer, Probability Measures on Locally Compact Groups, Springer, page 12",
          "text": "A locally compact group G is called Lie projective if it is the projective limit #x5C;textstyle#x5C;underset#x7B;#x5C;leftarrow#x7D;#x7B;#x5C;lim#x7D;#x5F;#x7B;#x5C;alpha#x5C;in#x5C;mathbb#x7B;A!!!G_α of Lie groups G_α:=G/K_α with a descending family (K_α)_(α∈𝔸) of compact normal subgroups K_α of G satisfying ∩_(α∈𝔸)K_α=e.}}",
          "type": "quotation"
        },
        {
          "ref": "2013, Eberhard Kaniuth, Keith F. Taylor, Induced Representations of Locally Compact Groups, Cambridge University Press, page 269",
          "text": "Let G be a locally compact group and H a closed subgroup of G, and suppose that #x5C;pi and #x5C;tau are irreducible representations of G and H, respectively.",
          "type": "quotation"
        }
      ],
      "glosses": [
        "A topological group whose underlying topology is both locally compact and Hausdorff."
      ],
      "hyponyms": [
        {
          "word": "Lie group"
        }
      ],
      "id": "en-locally_compact_group-en-noun-Yaenmz-x",
      "links": [
        [
          "topology",
          "topology"
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        [
          "topological group",
          "topological group"
        ],
        [
          "locally compact",
          "locally compact"
        ],
        [
          "Hausdorff",
          "Hausdorff"
        ]
      ],
      "raw_glosses": [
        "(topology) A topological group whose underlying topology is both locally compact and Hausdorff."
      ],
      "topics": [
        "mathematics",
        "sciences",
        "topology"
      ],
      "translations": [
        {
          "code": "de",
          "lang": "German",
          "sense": "topological group whose topology is locally compact and Hausdorff",
          "tags": [
            "feminine"
          ],
          "word": "lokalkompakte Gruppe"
        }
      ],
      "wikipedia": [
        "locally compact group"
      ]
    }
  ],
  "word": "locally compact group"
}

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    {
      "word": "Lie group"
    }
  ],
  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
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    {
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        {
          "ref": "1988, J. M. G. Fell, R. S. Doran, Representations of *-Algebras, Locally Compact Groups, and Banach *-Algebraic Bundles, Volume 1, Academic Press, page 63",
          "text": "Indeed, if there is one property of locally compact groups more responsible than any other for the rich positive content of their representation theory, it is their possession of left-invariant and right-invariant (Haar) measures. The connection between the representation theory of a locally compact group and that of general Banach algebras proceeds directly from Lebesgue integration with respect to Haar measure on the group.",
          "type": "quotation"
        },
        {
          "ref": "2012, H. Heyer, Probability Measures on Locally Compact Groups, Springer, page 12",
          "text": "A locally compact group G is called Lie projective if it is the projective limit #x5C;textstyle#x5C;underset#x7B;#x5C;leftarrow#x7D;#x7B;#x5C;lim#x7D;#x5F;#x7B;#x5C;alpha#x5C;in#x5C;mathbb#x7B;A!!!G_α of Lie groups G_α:=G/K_α with a descending family (K_α)_(α∈𝔸) of compact normal subgroups K_α of G satisfying ∩_(α∈𝔸)K_α=e.}}",
          "type": "quotation"
        },
        {
          "ref": "2013, Eberhard Kaniuth, Keith F. Taylor, Induced Representations of Locally Compact Groups, Cambridge University Press, page 269",
          "text": "Let G be a locally compact group and H a closed subgroup of G, and suppose that #x5C;pi and #x5C;tau are irreducible representations of G and H, respectively.",
          "type": "quotation"
        }
      ],
      "glosses": [
        "A topological group whose underlying topology is both locally compact and Hausdorff."
      ],
      "links": [
        [
          "topology",
          "topology"
        ],
        [
          "topological group",
          "topological group"
        ],
        [
          "locally compact",
          "locally compact"
        ],
        [
          "Hausdorff",
          "Hausdorff"
        ]
      ],
      "raw_glosses": [
        "(topology) A topological group whose underlying topology is both locally compact and Hausdorff."
      ],
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        "mathematics",
        "sciences",
        "topology"
      ],
      "wikipedia": [
        "locally compact group"
      ]
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  "translations": [
    {
      "code": "de",
      "lang": "German",
      "sense": "topological group whose topology is locally compact and Hausdorff",
      "tags": [
        "feminine"
      ],
      "word": "lokalkompakte Gruppe"
    }
  ],
  "word": "locally compact group"
}