Hilbert manifold

Summary

A Hilbert manifold is a Banach manifold whose charts take values in Hilbert spaces. Inner products give partitions of unity, orthogonal complements and Riemannian metrics, so much of finite-dimensional differential topology carries over. Spaces of paths and loops of Sobolev class $H^1$ in a finite-dimensional manifold are Hilbert manifolds. This makes them the natural setting for Morse theory in infinite dimensions. Take an infinite-dimensional separable model and a separable metrizable manifold. Then the open embedding theorem shows that it is an open subset of its model space, up to diffeomorphism.