---
tag: ZR09S9R3AW
kind: concept
name: Locally compact abelian group
msc: [22B05, 43A40]
specializes: [AHV05ZBCYW]
sources:
  - url: https://en.wikipedia.org/wiki/Locally_compact_abelian_group
same_as:
  wikidata: Q65122103
  wikipedia: Locally compact abelian group
---

## Summary

A locally compact abelian group, or LCA group, is a commutative topological group whose topology is Hausdorff and locally compact.
These groups are the natural setting of Fourier analysis.
Each LCA group has a Haar measure and a dual group of characters, and Pontryagin duality says that the group is recovered from its dual.

## Explanations

### L1 human

Think of a group where you can add elements, the order does not matter, and there is a notion of nearness with small compact neighbourhoods.
The real line $\R$ with addition is the basic picture: every point has a closed bounded interval around it.
The integers $\Z$ with the discrete topology, the circle group and every finite abelian group are also examples.
On all of these, one can integrate and expand functions into waves, just as Fourier series expand functions on the circle.