Banach–Lie group

Summary

A Banach–Lie group is a group that is also a smooth Banach manifold in which multiplication and inversion are smooth. It generalises a finite-dimensional Lie group 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.