Inmathematics, aBanach manifold is amanifold modeled onBanach spaces. Thus it is atopological space in which each point has aneighbourhoodhomeomorphic to anopen set in 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 manifold is a special case of a Banach manifold in which the manifold is locally modeled onHilbert spaces.

Let${\displaystyle X}$ be aset. Anatlas of class${\displaystyle C^r,}$${\displaystyle r\ge 0,}$ on${\displaystyle X}$ is a collection of pairs (calledcharts)${\displaystyle \left(U_i,{\phi }_i\right),}$${\displaystyle i\in I,}$ such that

1. each${\displaystyle U_i}$ is asubset of${\displaystyle X}$ and theunion of the${\displaystyle U_i}$ is the whole of${\displaystyle X}$;
2. each${\displaystyle {\phi }_i}$ is abijection from${\displaystyle U_i}$ onto anopen subset${\displaystyle {\phi }_i\left(U_i\right)}$ of some Banach space${\displaystyle E_i,}$ and for any indices${\displaystyle i\text{ and }j,}$${\displaystyle {\phi }_i\left(U_i\cap U_j\right)}$ is open in${\displaystyle E_i;}$
3. the crossover map$${\displaystyle {\phi }_j\circ {\phi }_i^{-1}:{\phi }_i\left(U_i\cap U_j\right)\to {\phi }_j\left(U_i\cap U_j\right)}$$ is an${\displaystyle r}$-times continuously differentiable function for every${\displaystyle i,j\in I;}$ that is, the${\displaystyle r}$thFréchet derivative$${\displaystyle {\mathrm{d}}^r\left({\phi }_j\circ {\phi }_i^{-1}\right):{\phi }_i\left(U_i\cap U_j\right)\to \mathrm{L}\mathrm{i}\mathrm{n}\left(E_i^r;E_j\right)}$$ exists and is acontinuous function with respect to the${\displaystyle E_i}$-normtopology on subsets of${\displaystyle E_i}$ and theoperator norm topology on${\displaystyle \mathrm{Lin}\left(E_i^r;E_j\right).}$

One can then show that there is a uniquetopology on${\displaystyle X}$ such that each${\displaystyle U_i}$ is open and each${\displaystyle {\phi }_i}$ 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${\displaystyle E_i}$ are equal to the same space${\displaystyle E,}$ the atlas is called an${\displaystyle E}$-atlas. However, it is nota priori necessary that the Banach spaces${\displaystyle E_i}$ be the same space, or evenisomorphic astopological vector spaces. However, if two charts${\displaystyle \left(U_i,{\phi }_i\right)}$ and${\displaystyle \left(U_j,{\phi }_j\right)}$ are such that${\displaystyle U_i}$ and${\displaystyle U_j}$ have a non-emptyintersection, a quick examination of thederivative of the crossover map$${\displaystyle {\phi }_j\circ {\phi }_i^{-1}:{\phi }_i\left(U_i\cap U_j\right)\to {\phi }_j\left(U_i\cap U_j\right)}$$shows that${\displaystyle E_i}$ and${\displaystyle E_j}$ must indeed be isomorphic as topological vector spaces. Furthermore, the set of points${\displaystyle x\in X}$ for which there is a chart${\displaystyle \left(U_i,{\phi }_i\right)}$ with${\displaystyle x}$ in${\displaystyle U_i}$ and${\displaystyle E_i}$ isomorphic to a given Banach space${\displaystyle E}$ is both open andclosed. Hence, one can without loss of generality assume that, on eachconnected component of${\displaystyle X,}$ the atlas is an${\displaystyle E}$-atlas for some fixed${\displaystyle E.}$

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

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

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

Similarly, if${\displaystyle U}$ is an open subset of some Banach space then${\displaystyle U}$ is a Banach manifold. (See theclassification theorem below.)

It is by no means true that a finite-dimensional manifold of dimension${\displaystyle n}$ isglobally homeomorphic to${\displaystyle {\mathbb{R}}^n,}$ or even an open subset of${\displaystyle {\mathbb{R}}^n.}$ However, in an infinite-dimensional setting, it is possible to classify "well-behaved" Banach manifolds up to homeomorphism quite nicely. A 1969 theorem ofDavid Henderson^[1] states that every infinite-dimensional,separable,metric Banach manifold${\displaystyle X}$ can beembedded as an open subset of the infinite-dimensional, separable Hilbert space,${\displaystyle H}$ (up to linear isomorphism, there is only one such space, usually identified with${\displaystyle {\ell }^2}$). 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${\displaystyle X.}$ Thus, in the infinite-dimensional, separable, metric case, the "only" Banach manifolds are the open subsets of Hilbert space.

• 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

• Abraham, Ralph; Marsden, J. E.; Ratiu, Tudor (1988).Manifolds, Tensor Analysis, and Applications. New York: Springer.ISBN 0-387-96790-7.
• 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.ISSN 0273-0979.S2CID 34049979.
• 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.ISSN 0002-9947.
• 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.MR 0247634.
• 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.

• Banach spaces
• Differential geometry
• Generalized manifolds
• Manifolds
• Nonlinear functional analysis
• Structures on manifolds

• Articles with short description
• Short description is different from Wikidata
• Pages displaying wikidata descriptions as a fallback via Module:Annotated link