atlases on A -manifold structure on is then defined to be a choice of equivalence class of atlases on of class If all the Banach spaces are isomorphic as topological vector spaces (which is guaranteed to be the case if is connected), then an equivalent atlas can be found for which they are all equal to some Banach space is then called an -manifold, or one says that is modeled on Examples If is a Banach space, then is a Banach manifold with an atlas containing a single, globally-defined chart (the identity map). Similarly, if is an open subset of some Banach space then is a Banach manifold. (See the classification theorem below.) Classification up to homeomorphism It is by no means true that a finite-dimensional manifold of dimension is homeomorphic to or even an open subset of However, in an infinite-dimensional setting, it is possible to classify "well-behaved" Banach manifolds up to homeomorphism quite nicely. A 1969 theorem of David Henderson states that every infinite-dimensional, separable, metric Banach manifold can be embedded as an open subset of the infinite-dimensional, separable Hilbert space, (up to linear isomorphism, there is only one such space, usually identified with ). In fact, Henderson's result is stronger: the same conclusion holds for any metric manifold modeled on a separable infinite-dimensional Fréchet space. The embedding homeomorphism can be used as a global chart for Thus, in the infinite-dimensional, separable, metric case, the "only" Banach manifolds are the open subsets of Hilbert space. See also References Banach spaces Differential geometry Generalized manifolds Manifolds Nonlinear functional analysis Structures on manifolds