nLab Banach manifold

Idea

A notion of infinite-dimensional manifold. A Banach manifold is a manifold modelled on Banach spaces. By default, transition maps are taken to be smooth.

Properties

As absolute neighbourhood retracts

Example

Every paracompact Banach manifold is an absolute neighbourhood retract.

By Palais 1966, Cor. to Thm. 5 on p. 3.

Embedding into the category of diffeological spaces

The category of smooth Banach manifolds has a full and faithful functor into the category of diffeological spaces. In terms of Chen smooth spaces this was observed in (Hain). For more see at Fréchet manifold – Relation to diffeological spaces.

Related concepts

• Hilbert manifold

• ILH manifold

• Frechet manifold

• convenient manifold

References

For general references see at infinite-dimensional manifold.

Aspects of the homotopy theory of Banach manifolds:

• Richard S. Palais, Homotopy theory of infinite dimensional manifolds, Topology 5 1 (1966) 1-16 (doi:10.1016/0040-9383(66)90002-4)

The full subcategory embedding into the category of diffeological spaces:

• Richard Hain, A characterization of smooth functions defined on a Banach space, Proc. Amer. Math. Soc. 77 (1979), 63-67 (web, pdf)