Smooth Banach manifold

A smooth Banach manifold modeled on a real Banach space $E$ is a Hausdorff space $M$ with a maximal atlas of homeomorphisms

\phi_i:U_i\longrightarrow V_i\subseteq E,

where the $U_i$ cover $M$ , the $V_i$ are open, and all transition maps $\phi_j\circ\phi_i^{-1}$ are $C^\infty$ . Here $C^\infty$ means that all iterated Fréchet derivatives exist and depend continuously on the point in the corresponding operator norms.

Smooth maps and Hilbert-valued orbit maps

A map between Banach manifolds is smooth when its expressions in charts are smooth. A Hilbert space is itself a Banach manifold. Thus a map from a finite-dimensional Lie group to a Hilbert space has an unambiguous notion of norm smoothness, even when its range is infinite-dimensional.

This is the smoothness used for the orbit maps defining smooth vectors. It does not ask that an operator-valued representation be smooth in operator norm.

Conventions

When $E=\mathbb R^n$ , this recovers the local chart definition of a finite-dimensional smooth manifold. Global countability and paracompactness assumptions are imposed separately here; neither is part of the displayed local definition. Complex Banach spaces may be used as real model spaces; holomorphic transition maps are an additional condition.

References

Karl-Hermann Neeb, On Differentiable Vectors for Representations of Infinite Dimensional Lie Groups. §2, Definition 2.1 and the ensuing manifold construction, specialized to Banach spaces.