Math 2.0

Topological groups, Lie groups Definition

Locally compact group

Builds on Topological groupLocally compact spaceHausdorff space

Definition

A locally compact group is a topological group GG such that:

  1. the topology of GG is Hausdorff;
  2. the topology of GG is locally compact, that is, every point of GG has a compact neighbourhood.

Remarks

Example: (ℝn,+)(\R^n, +) is a locally compact group, since the closed ball of radius 11 around xx is a compact neighbourhood of xx by the Heine–Borel theorem. Every discrete group is locally compact, since {x}\set{x} is a compact neighbourhood of xx.

Non-example: the additive group of an infinite-dimensional Hilbert space HH is a Hausdorff topological group but is not locally compact. Suppose 00 had a compact neighbourhood KK. Then KK contains a closed ball of some radius r>0r > 0, and this ball is compact as a closed subset of KK. For an orthonormal sequence (ek)(e_k), the points rekr e_k lie in this ball. They have no convergent subsequence, a contradiction, since ∥rej−rek∥=r2\norm{r e_j - r e_k} = r\sqrt{2} for j≠kj \ne k.

By homogeneity it is enough that the identity has a compact neighbourhood. Some authors omit the Hausdorff condition; a locally compact group in that wider sense becomes Hausdorff after dividing by the closure of the identity.

Reading path

To understand this page from the ground up, read these 35 pages in order. The foundations come first.

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Element, definition, no evidence levelElementdefinitionProposition, concept, no evidence levelPropositionconceptReal numbers, object, no evidence levelReal numbersobjectTopological group, concept, no evidence levelTopological groupconceptSet, definition, no evidence levelSetdefinitionConjunction, definition, no evidence levelConjunctiondefinitionDisjunction, definition, no evidence levelDisjunctiondefinitionImplication, definition, no evidence levelImplicationdefinitionPredicate, definition, no evidence levelPredicatedefinitionEmpty set, object, no evidence levelEmpty setobjectFamily of sets, definition, no evidence levelFamily of setsdefinitionOrdered pair, definition, no evidence levelOrdered pairdefinitionIntersection, definition, no evidence levelIntersectiondefinitionUnion, definition, no evidence levelUniondefinitionExistential quantifier, definition, no evidence levelExistential quantifierdefinitionUniversal quantifier, definition, no evidence levelUniversal quantifierdefinitionCartesian product, definition, no evidence levelCartesian productdefinitionSubset, definition, no evidence levelSubsetdefinitionUnion of a family of sets, definition, no evidence levelUnion of a family of …definitionUniqueness quantifier, definition, no evidence levelUniqueness quantifierdefinitionBinary relation, definition, no evidence levelBinary relationdefinitionTopological space, definition, no evidence levelTopological spacedefinitionFunction, definition, no evidence levelFunctiondefinitionSubspace topology, definition, no evidence levelSubspace topologydefinitionMetric space, definition, no evidence levelMetric spacedefinitionOpen ball, definition, no evidence levelOpen balldefinitionMetric topology, definition, no evidence levelMetric topologydefinitionOpen set, definition, no evidence levelOpen setdefinitionHausdorff space, definition, no evidence levelHausdorff spacedefinitionOpen cover, definition, no evidence levelOpen coverdefinitionOpen sets characterised by neighbourhoods, statement, no evidence levelOpen sets characteris…statementCompact space, definition, no evidence levelCompact spacedefinitionCompact set, definition, no evidence levelCompact setdefinitionLocally compact space, definition, no evidence levelLocally compact spacedefinitionLocally compact group, definition, no evidence levelLocally compact groupdefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Element definition
  2. Proposition concept
  3. Real numbers object
  4. Topological group concept
  5. Set definition
  6. Conjunction definition
  7. Disjunction definition
  8. Implication definition
  9. Predicate definition
  10. Empty set object
  11. Family of sets definition
  12. Ordered pair definition
  13. Intersection definition
  14. Union definition
  15. Existential quantifier definition
  16. Universal quantifier definition
  17. Cartesian product definition
  18. Subset definition
  19. Union of a family of sets definition
  20. Uniqueness quantifier definition
  21. Binary relation definition
  22. Topological space definition
  23. Function definition
  24. Subspace topology definition
  25. Metric space definition
  26. Open ball definition
  27. Metric topology definition
  28. Open set definition
  29. Hausdorff space definition
  30. Open cover definition
  31. Open sets characterised by neighbourhoods statement
  32. Compact space definition
  33. Compact set definition
  34. Locally compact space definition
  35. Locally compact group definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Locally_compact_group
About this page
Tag20YERDAS5W
URIm2://20YERDAS5W
Kinddefinition
WritingW0 stub
MSC22D05, 22A05
DefinesLocally compact group
UsesTopological group, Locally compact space, Hausdorff space

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