# Functional Analysis/Harmonic Analysis/Locally Compact Groups

## Introduction

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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.

## Preliminaries

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 ${\displaystyle l^{2}(\mathbb {N} )}$  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 ${\displaystyle H\subset G}$  be an open subgroup of ${\displaystyle G}$ . Choose a set ${\displaystyle \{y_{j}\in G|\ j\in J\}}$ , one ${\displaystyle y_{j}}$  for each class in ${\displaystyle G/H}$ , but choosing ${\displaystyle e}$  for the class of ${\displaystyle H}$ . We then have the disjoint union ${\displaystyle G=\sqcup _{j\in J}y_{j}H}$ . Since left multiplication by a given element ${\displaystyle y_{j}}$  is a homeomorphism between ${\displaystyle H}$  and ${\displaystyle y_{j}H}$ , we have that each such set is open in ${\displaystyle G}$ . Therefore the complement of ${\displaystyle H=eH}$  is open in ${\displaystyle G}$  and therefore ${\displaystyle H}$  is also closed.

If now ${\displaystyle H\subset G}$  be an closed subgroup of ${\displaystyle G}$ , let ${\displaystyle x\in H}$ . There exists a compact neighborhood ${\displaystyle X}$  of ${\displaystyle x}$  in ${\displaystyle G}$ . But then the intersection ${\displaystyle X\cap H}$  is a compact neighborhood of ${\displaystyle x}$  in ${\displaystyle H}$ . 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 ${\displaystyle G}$  be a topological group. In order for ${\displaystyle G}$  to be locally compact it is necessary and sufficient that the neutral element ${\displaystyle e}$  possesses a compact neighborhood.

Proof: Indeed, if ${\displaystyle X}$  is a compact neighborhood of ${\displaystyle e}$ , then ${\displaystyle Xx}$  is a compact neighborhood of ${\displaystyle x}$  for any ${\displaystyle x\in X}$ , since ${\displaystyle Xx=R_{x}(X)}$  is the image of a continuous map by exercise (ref) (left and right multiplication maps). QED.

## Appendices

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.