Banach manifold

Inmathematics,aBanach manifoldis amanifoldmodeled onBanach spaces.Thus it is atopological spacein which each point has aneighbourhoodhomeomorphicto anopen setin a Banach space (a more involved and formal definition is given below). Banach manifolds are one possibility of extending manifolds toinfinitedimensions.

A further generalisation is toFréchet manifolds,replacing Banach spaces byFréchet spaces.On the other hand, aHilbert manifoldis a special case of a Banach manifold in which the manifold is locally modeled onHilbert spaces.

Definition

Let$X$be aset.Anatlasof class$C^{r},$$r\geq 0,$on$X$is a collection of pairs (calledcharts)$\left(U_{i},\varphi _{i}\right),$$i\in I,$such that

1. each$U_{i}$is asubsetof$X$and theunionof the$U_{i}$is the whole of$X$;
2. each$\varphi _{i}$is abijectionfrom$U_{i}$onto anopen subset$\varphi _{i}\left(U_{i}\right)$of some Banach space$E_{i},$and for any indices$i{\text{ and }}j,$$\varphi _{i}\left(U_{i}\cap U_{j}\right)$is open in$E_{i};$
3. the crossover map$$\varphi _{j}\circ \varphi _{i}^{-1}:\varphi _{i}\left(U_{i}\cap U_{j}\right)\to \varphi _{j}\left(U_{i}\cap U_{j}\right)$$is an$r$-times continuously differentiablefunction for every$i,j\in I;$that is, the$r$thFréchet derivative$$\mathrm {d} ^{r}\left(\varphi _{j}\circ \varphi _{i}^{-1}\right):\varphi _{i}\left(U_{i}\cap U_{j}\right)\to \mathrm {Lin} \left(E_{i}^{r};E_{j}\right)$$exists and is acontinuous functionwith respect to the$E_{i}$-normtopologyon subsets of$E_{i}$and theoperator normtopology on$\operatorname {Lin} \left(E_{i}^{r};E_{j}\right).$

One can then show that there is a uniquetopologyon$X$such that each$U_{i}$is open and each$\varphi _{i}$is ahomeomorphism.Very often, this topological space is assumed to be aHausdorff space,but this is not necessary from the point of view of the formal definition.

If all the Banach spaces$E_{i}$are equal to the same space$E,$the atlas is called an$E$-atlas.However, it is nota priorinecessary that the Banach spaces$E_{i}$be the same space, or evenisomorphicastopological vector spaces.However, if two charts$\left(U_{i},\varphi _{i}\right)$and$\left(U_{j},\varphi _{j}\right)$are such that$U_{i}$and$U_{j}$have a non-emptyintersection,a quick examination of thederivativeof the crossover map $$\varphi _{j}\circ \varphi _{i}^{-1}:\varphi _{i}\left(U_{i}\cap U_{j}\right)\to \varphi _{j}\left(U_{i}\cap U_{j}\right)$$ shows that$E_{i}$and$E_{j}$must indeed be isomorphic as topological vector spaces. Furthermore, the set of points$x\in X$for which there is a chart$\left(U_{i},\varphi _{i}\right)$with$x$in$U_{i}$and$E_{i}$isomorphic to a given Banach space$E$is both open andclosed.Hence, one can without loss of generality assume that, on eachconnected componentof$X,$the atlas is an$E$-atlas for some fixed$E.$

A new chart$(U,\varphi )$is calledcompatiblewith a given atlas$\left\{\left(U_{i},\varphi _{i}\right):i\in I\right\}$if the crossover map $$\varphi _{i}\circ \varphi ^{-1}:\varphi \left(U\cap U_{i}\right)\to \varphi _{i}\left(U\cap U_{i}\right)$$ is an$r$-times continuously differentiable function for every$i\in I.$Two atlases are called compatible if every chart in one is compatible with the other atlas. Compatibility defines anequivalence relationon the class of all possible atlases on$X.$

A$C^{r}$-manifoldstructure on$X$is then defined to be a choice of equivalence class of atlases on$X$of class$C^{r}.$If all the Banach spaces$E_{i}$are isomorphic as topological vector spaces (which is guaranteed to be the case if$X$isconnected), then an equivalent atlas can be found for which they are all equal to some Banach space$E.$$X$is then called an$E$-manifold,or one says that$X$ismodeledon$E.$

Examples

Every Banach space can be canonically identified as a Banach manifold. If$(X,\|\,\cdot \,\|)$is a Banach space, then$X$is a Banach manifold with an atlas containing a single, globally-defined chart (theidentity map).

Similarly, if$U$is an open subset of some Banach space then$U$is a Banach manifold. (See theclassification theorembelow.)

Classification up to homeomorphism

It is by no means true that a finite-dimensional manifold of dimension$n$isgloballyhomeomorphic to$\mathbb {R} ^{n},$or even an open subset of$\mathbb {R} ^{n}.$However, in an infinite-dimensional setting, it is possible to classify "well-behaved"Banach manifolds up to homeomorphism quite nicely. A 1969 theorem ofDavid Hendersonstates that every infinite-dimensional,separable,metricBanach manifold$X$can beembeddedas an open subset of the infinite-dimensional, separable Hilbert space,$H$(up to linear isomorphism, there is only one such space, usually identified with$\ell ^{2}$). In fact, Henderson's result is stronger: the same conclusion holds for any metric manifold modeled on a separable infinite-dimensionalFréchet space.

The embedding homeomorphism can be used as a global chart for$X.$Thus, in the infinite-dimensional, separable, metric case, the "only" Banach manifolds are the open subsets of Hilbert space.

See also

• Banach bundle– vector bundle whose fibres form Banach spacesPages displaying wikidata descriptions as a fallback
• Differentiation in Fréchet spaces
• Finsler manifold– Generalization of Riemannian manifolds
• Fréchet manifold– topological space modeled on a Fréchet space in much the same way as a manifold is modeled on a Euclidean spacePages displaying wikidata descriptions as a fallback
• Global analysis– which uses Banach manifolds and other kinds of infinite-dimensional manifolds
• Hilbert manifold– Manifold modelled on Hilbert spaces

References

• Abraham, Ralph; Marsden, J. E.; Ratiu, Tudor (1988).Manifolds, Tensor Analysis, and Applications.New York: Springer.ISBN0-387-96790-7.
• Anderson, R. D. (1969)."Strongly negligible sets in Fréchet manifolds"(PDF).Bulletin of the American Mathematical Society.75(1). American Mathematical Society (AMS): 64–67.doi:10.1090/s0002-9904-1969-12146-4.ISSN0273-0979.S2CID34049979.
• Anderson, R. D.; Schori, R. (1969)."Factors of infinite-dimensional manifolds"(PDF).Transactions of the American Mathematical Society.142.American Mathematical Society (AMS): 315–330.doi:10.1090/s0002-9947-1969-0246327-5.ISSN0002-9947.
• Henderson, David W. (1969)."Infinite-dimensional manifolds are open subsets of Hilbert space".Bull. Amer. Math. Soc.75(4): 759–762.doi:10.1090/S0002-9904-1969-12276-7.MR0247634.
• Lang, Serge(1972).Differential manifolds.Reading, Mass.–London–Don Mills, Ont.: Addison-Wesley Publishing Co., Inc.
• Zeidler, Eberhard (1997).Nonlinear functional analysis and its Applications. Vol.4.Springer-Verlag New York Inc.