Functional Analysis/Harmonic Analysis/Locally Compact Groups

IntroductionEdit

In this section we define the most well-known class of topological groups, namely locally compact groups. This class includes compact groups which in turn includes all finite groups, finite-dimensional Lie groups, etc.


Exercises 0% developedEdit

Preliminaries 0% developedEdit

Definition 9.2.1: A locally compact group is a topological group whose underlying topological space is locally compact.

Examples:

  1. All compact, and therefore all finite groups are locally compact.
  2. A discrete group is always locally compact.
  3. Any finite-dimensional vector space is a locally compact group (equipped with addition).


The Hilbert space   is not locally compact in the norm topology.

Proposition: An open subgroup of a locally compact group is always closed. A closed subgroup of a locally compact group is locally compact.

Proof: Indeed, let   be an open subgroup of  . Choose a set  , one   for each class in  , but choosing   for the class of  . We then have the disjoint union  . Since left multiplication by a given element   is a homeomorphism between   and  , we have that each such set is open in  . Therefore the complement of   is open in   and therefore   is also closed.

If now   be an closed subgroup of  , let  . There exists a compact neighborhood   of   in  . But then the intersection   is a compact neighborhood of   in  . QED.

Combining the statements in the last proposition we conclude that an open subgroup of a locally compact group is also locally compact.

Proposition: Let   be a topological group. In order for   to be locally compact it is necessary and sufficient that the neutral element   possesses a compact neighborhood.

Proof: Indeed, if   is a compact neighborhood of  , then   is a compact neighborhood of   for any  , since   is the image of a continuous map by exercise (ref) (left and right multiplication maps). QED.




Appendices 0% developedEdit

Here, you will find a list of unsorted chapters. Some of them listed here are highly advanced topics, while others are tools to aid you on your mathematical journey. Since this is the last heading for the wikibook, the necessary book endings are also located here.