Banach manifold
A Banach manifold is a topological space in which there is a neighborhood for every point that is homeomorphic to a Banach space .
definition
The definition of a Banach manifold differs from that of a manifold only in that the cards
Have images in a (possibly infinite-dimensional) Banach space and the concatenated mapping
is r-times differentiable and therefore the r-th Fréchet derivative
exists and is a continuous function in relation to the standard topology on subsets of and the operator standard topology on .
Examples
If is a Banach space, then is a Banach manifold whose atlas contains a single map that is globally defined. Likewise, an open subset of a Banach space is a Banach manifold.
Classifications and Homeomorphisms
Although a finite- dimensional manifold to not globally homeomorphic or a subset of these is some Banach manifolds can be in an infinite-dimensional framework to classify homeomorphism. The mathematician David Henderson proved in 1969 that every infinite-dimensional, separable, metric Banach manifold can be embedded as an open subset in the infinite-dimensional, separable Hilbert space. The result is an even more general statement, which is that this is true for every metric manifold defined by maps in a separable Fréchet space .
Banach bundle
definition
Given a Banach manifold of the class with which the base space represents, a topological space as total space and a mapping . The fiber has the structure of a Banach space.
Let be an open cover of . For each there is a Banach space and a map
- ,
so that
- the mapping is a homeomorphism that commutes with the projection and for all the induced mapping on the fiber
is a continuous, invertible mapping and therefore an isomorphism in the category of topological vector spaces (in the context of a usual definition of a fiber bundle, this corresponds to a transition function).
- If and are two members of the open overlap, then the mapping is
a morphism. is the set of continuous linear mappings between two topological vector spaces and .
The family is called trivial cover for and the mappings are called local trivialization. This data determines a fiber bundle structure on the Banach manifold .
literature
- Eberhard Zeidler: Nonlinear functional analysis and its applications. Vol.4 . Springer-Verlag New York Inc., 1997.
Individual evidence
- ↑ David Henderson: Infinite-dimensional manifolds are open subsets of Hilbert space. Bull. Amer. Math. Soc. 75: 759-762 (1969).