---
tag: MC5CGVRFJM
kind: concept
name: Banach–Lie group
msc: ["22E65", 58B25]
specializes: [92700QAB5Y]
sources:
  - url: https://en.wikipedia.org/wiki/Lie_group
---

## Summary

A Banach–Lie group is a group that is also a smooth [[Banach manifold|9K15AMGBV9]] in which multiplication and inversion are smooth.
It generalises a finite-dimensional [[Lie group|7JBRGCVJRB]] to infinite dimension.
Its Lie algebra is a Banach–Lie algebra, and, unlike for more general infinite-dimensional groups, the exponential map is still a local diffeomorphism at $0$.
Basic examples are the group of invertible bounded operators on a Banach space and the unitary group of a Hilbert space with the norm topology.


## Explanations

### L1 human

A Lie group is a group whose elements form a smooth space, such as the rotations of space.
A Banach–Lie group is the same idea when the group has infinitely many directions.
For example, the invertible bounded operators on the Hilbert space $\ell^2$ form a group, and near the identity they look like a ball of the space of all bounded operators.
Many tools of finite-dimensional Lie theory, such as the exponential map, keep working in this setting.