Banach–Lie group - Knowlpedia

A Banach–Lie group is a group $G$ equipped with a smooth Banach-manifold structure such that

m:G\times G\longrightarrow G,\quad m(g,h)=gh, \qquad \operatorname{inv}:G\longrightarrow G,\quad g\longmapsto g^{-1}

are smooth. Its model space may be infinite-dimensional.

Tangent Lie algebra

The tangent space $\mathfrak g=T_eG$ is a Banach space. The bracket obtained from left-invariant vector fields is continuous and bilinear, and the exponential map is a local diffeomorphism near $0$ . Finite-dimensional Lie groups are special cases.

Example and scope

The additive group of a Banach space is a Banach–Lie group. The unitary group with operator-norm topology is another example.

A group with a weaker topology does not inherit this smooth structure merely because it has the same underlying elements. In particular, strong continuity of a unitary representation need not make it a smooth map into the norm-topology unitary group.

References

Karl-Hermann Neeb, On Differentiable Vectors for Representations of Infinite Dimensional Lie Groups. §2, definition of locally convex Lie groups and the Banach–Lie specialization.