Functional Analysis/Harmonic Analysis/Locally Compact Groups
Introduction[edit | edit source]
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 [edit | edit source]
Preliminaries [edit | edit source]
Definition 9.2.1: A locally compact group is a topological group whose underlying topological space is locally compact.
- All compact, and therefore all finite groups are locally compact.
- A discrete group is always locally compact.
- 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 [edit | edit source]
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.