---
tag: AHV05ZBCYW
kind: concept
name: Locally compact group
msc: ["22D05", "22A05"]
specializes: [7GKCGQ7215]
sources:
  - url: https://en.wikipedia.org/wiki/Locally_compact_group
same_as:
  wikidata: Q2147620
  wikipedia: Locally compact group
---

## Summary

A locally compact group is a Hausdorff [[topological group|92700QAB5Y]] whose topology is [[locally compact|0FE2SB8WVJ]].
These groups carry a [[Haar measure|VY0HXD3GC4]], an invariant way to measure and integrate, which makes harmonic analysis on them possible.
They include all Lie groups, all discrete groups, all compact groups and the groups of points of algebraic groups over local fields such as $\Q_p$.

## Explanations

### L1 human

A space is locally compact if every point has a small compact patch around it, like a closed and bounded interval around a point of the line.
The real line, the circle and the $p$-adic numbers all have this property, while an infinite-dimensional Hilbert space does not.
A locally compact group is a group whose space of elements is of this kind and whose operations are continuous.
For example, the group of invertible $n \times n$ real matrices is locally compact, since it is an open subset of $\R^{n^2}$.