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 . ${\displaystyle X}$ ${\displaystyle x\in {X}}$

definition

The definition of a Banach manifold differs from that of a manifold only in that the cards

${\displaystyle \varphi _{i}\colon U_{i}\to \varphi _{i}(U_{i})\subset E_{i}}$

Have images in a (possibly infinite-dimensional) Banach space and the concatenated mapping ${\displaystyle E_{i}}$

${\displaystyle \varphi _{j}\circ \varphi _{i}^{-1}\colon \varphi _{i}(U_{i}\cap U_{j})\to \varphi _{j}(U_{i}\cap U_{j})}$

is r-times differentiable and therefore the r-th Fréchet derivative

${\displaystyle \mathrm {d} ^{r}{\big (}\varphi _{j}\circ \varphi _{i}^{-1}{\big )}\colon \varphi _{i}(U_{i}\cap U_{j})\to \operatorname {Lin} {\big (}E_{i}^{r};E_{j}{\big )}}$

exists and is a continuous function in relation to the standard topology on subsets of and the operator standard topology on . ${\displaystyle E_{i}}$ ${\displaystyle E_{i}}$ ${\displaystyle \operatorname {Lin} {\big (}E_{i}^{r};E_{j}{\big )}}$

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. ${\displaystyle ({X},\left\|\cdot \right\|)}$ ${\displaystyle X}$ ${\displaystyle U}$

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 . ${\displaystyle n}$ ${\displaystyle \mathbb {R} ^{n}}$

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. ${\displaystyle M}$ ${\displaystyle C^{p}}$ ${\displaystyle p\geq 0}$ ${\displaystyle E}$ ${\displaystyle \pi :E\rightarrow M}$ ${\displaystyle \pi ^{-1}(x)=:E_{\text{X}}}$

Let be an open cover of . For each there is a Banach space and a map ${\displaystyle \{U_{i}|i\in I\}}$ ${\displaystyle M}$ ${\displaystyle i\in {I}}$ ${\displaystyle {X}_{i}}$ ${\displaystyle \tau _{i}}$

${\displaystyle \tau _{i}:\pi ^{-1}(U_{i})\to U_{i}\times X_{i}}$,

so that

• the mapping is a homeomorphism that commutes with the projection and for all the induced mapping on the fiber ${\displaystyle \tau _{i}}$ ${\displaystyle U_{i}}$ ${\displaystyle x\in {U}_{i}}$ ${\displaystyle \tau _{ix}}$ ${\displaystyle E_{X}}$

${\displaystyle \tau _{ix}:\pi ^{-1}(x)\to X_{i}}$

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 ${\displaystyle U_{i}}$ ${\displaystyle U_{j}}$

${\displaystyle U_{i}\cap U_{j}\to \mathrm {Lin} (X_{i};X_{j})}$

${\displaystyle x\mapsto (\tau _{j}\circ \tau _{i}^{-1})_{x}}$

a morphism. is the set of continuous linear mappings between two topological vector spaces and . ${\displaystyle Lin({{X};{Y}})}$ ${\displaystyle X}$ ${\displaystyle Y}$

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 . ${\displaystyle {(U_{i};\tau _{i})|i\in {I}}}$ ${\displaystyle \pi :E\rightarrow M}$ ${\displaystyle \tau _{i}}$ ${\displaystyle M}$

literature

• Eberhard Zeidler: Nonlinear functional analysis and its applications. Vol.4 . Springer-Verlag New York Inc., 1997.

Individual evidence

1. ↑ David Henderson: Infinite-dimensional manifolds are open subsets of Hilbert space. Bull. Amer. Math. Soc. 75: 759-762 (1969).