Locally compact group

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In mathematics, a locally compact group is a topological group G which is locally compact as a topological space. Locally compact groups are important because they have a natural measure called the Haar measure. This allows one to define integrals of Borel measurable functions on G.

Many of the results of finite group representation theory are proved by averaging over the group. These proofs can be carried over to locally compact groups by replacement of the average with the Haar integral. The resulting theory is a central part of harmonic analysis. The theory for locally compact abelian groups is described by Pontryagin duality, a generalized Fourier transform.

Examples and counterexamples

  • Any compact group is locally compact.
  • Any discrete group is locally compact. The theory of locally compact groups therefore encompasses the theory of ordinary groups since any group can be given the discrete topology.
  • Lie groups, which are locally Euclidean, are all locally compact groups.
  • A Hausdorff topological vector space is locally compact if and only if it is finite-dimensional.
  • The additive group of rational numbers Q is not locally compact if given the relative topology as a subset of the real numbers. It is locally compact if given the discrete topology.
  • The additive group of p-adic numbers Qp is locally compact for any prime number p.

Properties

By homogeneity, local compactness for a topological group need only be checked at the identity. That is, a group G is locally compact if and only if the identity element has a compact neighborhood. It follows that there is a local base of compact neighborhoods at every point.

Every closed subgroup of a locally compact group is locally compact. (The closure condition is necessary as the group of rationals demonstrates.) Conversely, every locally compact subgroup of a Hausdorff group is closed. Every quotient of a locally compact group is locally compact. The product of a family of locally compact groups is locally compact if and only if all but a finite number of factors are actually compact.

Topological groups are always completely regular as topological spaces. Locally compact groups have the stronger property of being normal.

Every locally compact group which is second-countable is metrizable as a topological group (i.e. can be given a left-invariant metric compatible with the topology) and complete.

See also

References

  • Folland, Gerald B. (1995), A Course in Abstract Harmonic Analysis, CRC Press, ISBN 978-0-8493-8490-5 .
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