Банах кеңістіктерінде модельделген манифольдтар

Manifold modeled on Banach spaces
In mathematics, a Banach manifold is a manifold modeled on Banach spaces. Thus it is a topological space in which each point has a neighbourhood homeomorphic to an open set in a Banach space (a more involved and formal definition is given below). Banach manifolds are one possibility of extending manifolds to infinite dimensions. A further generalisation is to Fréchet manifolds, replacing Banach spaces by Fréchet spaces. On the other hand, a Hilbert manifold is a special case of a Banach manifold in which the manifold is locally modeled on Hilbert spaces.

Every Banach space can be canonically identified as a Banach manifold. 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.)

It is by no means true that a finite dimensional manifold of dimension is globally 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.