Banach manifold

Short description: 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.

## Contents

* 1 Definition
* 2 Examples
* 3 Classification up to homeomorphism
* 4 See also
* 5 References

## Definition

Let X be a set. An atlas of class Cr, r≥0, on X is a collection of pairs (called charts) (Ui,φi), i∈I, such that

1. each Ui is a subset of X and the union of the Ui is the whole of X;
2. each φi is a bijection from Ui onto an open subset φi(Ui) of some Banach space Ei, and for any indices i and j, φi(Ui∩Uj) is open in Ei;
3. the crossover map φj∘φi−1:φi(Ui∩Uj)->φj(Ui∩Uj) is an r-times continuously differentiable function for every i,j∈I; that is, the rth Fréchet derivative dr(φj∘φi−1):φi(Ui∩Uj)->Lin(Eir;Ej) exists and is a continuous function with respect to the Ei-norm topology on subsets of Ei and the operator norm topology on Lin⁡(Eir;Ej).

One can then show that there is a unique topology on X such that each Ui is open and each φi is a homeomorphism. Very often, this topological space is assumed to be a Hausdorff space, but this is not necessary from the point of view of the formal definition.

If all the Banach spaces Ei are equal to the same space E, the atlas is called an E-atlas. However, it is not a priori necessary that the Banach spaces Ei be the same space, or even isomorphic as topological vector spaces. However, if two charts (Ui,φi) and (Uj,φj) are such that Ui and Uj have a non-empty intersection, a quick examination of the derivative of the crossover map φj∘φi−1:φi(Ui∩Uj)->φj(Ui∩Uj) shows that Ei and Ej must indeed be isomorphic as topological vector spaces. Furthermore, the set of points x∈X for which there is a chart (Ui,φi) with x in Ui and Ei isomorphic to a given Banach space E is both open and closed. Hence, one can without loss of generality assume that, on each connected component of X, the atlas is an E-atlas for some fixed E.

A new chart (U,φ) is called compatible with a given atlas {(Ui,φi):i∈I} if the crossover map φi∘φ−1:φ(U∩Ui)->φi(U∩Ui) is an r-times continuously differentiable function for every i∈I. Two atlases are called compatible if every chart in one is compatible with the other atlas. Compatibility defines an equivalence relation on the class of all possible atlases on X.

A Cr-manifold structure on X is then defined to be a choice of equivalence class of atlases on X of class Cr. If all the Banach spaces Ei are isomorphic as topological vector spaces (which is guaranteed to be the case if X is connected), 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 is modeled on E.

## Examples

Every Banach space can be canonically identified as a Banach manifold. If (X,‖⋅‖) is a Banach space, then X is a Banach manifold with an atlas containing a single, globally defined chart (the identity map).

Similarly, if U is an open subset of some Banach space then U is a Banach manifold. (See the classification theorem below.)

## Classification up to homeomorphism

It is by no means true that a finite-dimensional manifold of dimension n is globally homeomorphic to ℝn, or even an open subset of ℝn. 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[1] states that every infinite-dimensional, separable, metric Banach manifold X can be embedded as an open subset of the infinite-dimensional, separable Hilbert space, H (up to linear isomorphism, there is only one such space, usually identified with ℓ2). 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 X. Thus, in the infinite-dimensional, separable, metric case, the "only" Banach manifolds are the open subsets of Hilbert space.

## See also

* Banach bundle
* Differentiation in Fréchet spaces
* Finsler manifold - Generalization of Riemannian manifolds
* Fréchet manifold
* Global analysis – which uses Banach manifolds and other kinds of infinite-dimensional manifolds
* Hilbert manifold - Manifold modelled on Hilbert spaces

## References

1. ↑ Henderson 1969.

* 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". Bulletin of the American Mathematical Society (American Mathematical Society (AMS)) 75 (1): 64–67. doi:10.1090/s0002-9904-1969-12146-4. ISSN 0273-0979. https://www.ams.org/journals/proc/1969-023-03/S0002-9939-1969-0248883-5/S0002-9939-1969-0248883-5.pdf.
* Anderson, R. D.; Schori, R. (1969). "Factors of infinite-dimensional manifolds". Transactions of the American Mathematical Society (American Mathematical Society (AMS)) 142: 315–330. doi:10.1090/s0002-9947-1969-0246327-5. ISSN 0002-9947. https://www.ams.org/journals/tran/1969-142-00/S0002-9947-1969-0246327-5/S0002-9947-1969-0246327-5.pdf.
* 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.
* 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..

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Manifolds (Glossary, List, Category)

Basic concepts|

* Topological manifold
* Atlas
* Differentiable/Smooth manifold
* Differential structure
* Smooth atlas
* Submanifold
* Riemannian manifold
* Smooth map
* Submersion
* Pushforward
* Tangent space
* Differential form
* Vector field

Main theorems (list)|

* Atiyah–Singer index
* Darboux's
* De Rham's
* Frobenius
* Generalized Stokes
* Hopf–Rinow
* Noether's
* Sard's
* Whitney embedding

Maps|

* Curve
* Diffeomorphism
* Local
* Geodesic
* Exponential map
* in Lie theory
* Foliation
* Immersion
* Integral curve
* Lie derivative
* Section
* Submersion

Types of
manifolds|

* Calabi–Yau
* Closed
* Collapsing
* Complete
* (Almost) Complex
* (Almost) Contact
* Einstein
* Fibered
* Finsler
* (Almost, Ricci-) Flat
* G-structure
* Hadamard
* Hermitian
* Hyperbolic
* (Hyper) Kähler
* Kenmotsu
* Lie group
* Lie algebra
* Manifold with boundary
* Nilmanifold
* Oriented
* Parallelizable
* Poisson
* Prime
* Quaternionic
* Hypercomplex
* (Pseudo-, Sub-) Riemannian
* Rizza
* Sasakian
* Stein
* (Almost) Symplectic
* Tame

Tensors|

| Vectors|

* Distribution
* Lie bracket
* Pushforward
* Tangent space
* bundle
* Torsion
* Vector field
* Vector flow

|

Covectors|

* Closed/Exact
* Covariant derivative
* Cotangent space
* bundle
* De Rham cohomology
* Differential form
* Complex
* Vector-valued
* One-form
* Exterior derivative
* Interior product
* Pullback
* Ricci curvature
* flow
* Riemann curvature tensor
* Tensor field
* density
* Volume form
* Wedge product

Bundles|

* Adjoint
* Affine
* Associated
* Cotangent
* Dual
* Fiber
* (Co-) Fibration
* Jet
* Lie algebra
* (Stable) Normal
* Principal
* Spinor
* Subbundle
* Tangent
* Tensor
* Vector

Connections|

* Affine
* Cartan
* Ehresmann
* Form
* Generalized
* Koszul
* Levi-Civita
* Principal
* Vector
* Parallel transport

Related|

* Classification of manifolds
* Gauge theory
* History
* Morse theory
* Moving frame
* Singularity theory

Generalizations|

* Banach
* Diffeology
* Diffiety
* Fréchet
* Hilbert
* K-theory
* Non-Hausdorff
* Orbifold
* Secondary calculus
* over commutative algebras
* Sheaf
* Stratifold
* Supermanifold
* Stratified space

Template:Analysis in topological vector spaces Template:Banach spaces

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Functional analysis (topics)

Topological vector spaces|

* Asplund
* Banach (list)
* Banach lattice
* Barrelled
* Bornological
* Brauner
* F-space
* Fréchet (tame)
* Hilbert (Inner product space
* Polarization identity)
* LF-space
* Locally convex (Seminorms/Minkowski functionals)
* Mackey
* Montel
* Nuclear
* Normed (norm)
* Quasinormed
* Reflexive
* Riesz
* Smith
* Stereotype
* Strictly convex
* Webbed
* Topological tensor product (of Hilbert spaces)

Topologies of function spaces|

* Dual
* Dual space (Dual norm)
* Operator
* Ultraweak
* Weak (polar
* operator)
* Mackey
* Strong (polar
* operator)
* Ultrastrong
* Uniform convergence

Linear operators|

* Adjoint
* Bilinear (form
* operator
* sesquilinear)
* (Un)Bounded
* Closed
* Compact (on Hilbert spaces)
* (Dis)Continuous
* Densely defined
* Fredholm
* Hilbert–Schmidt
* Functionals (positive)
* Normal
* Nuclear
* Self-adjoint
* Strictly singular
* Trace class
* Transpose
* Unitary

Operator theory|

* Banach algebras
* C*-algebras
* Spectrum (C*-algebra
* radius)
* Spectral theory (of ODEs
* Spectral theorem)
* Polar decomposition
* Singular value decomposition

Theorems|

* Banach–Alaoglu
* Banach–Mazur
* Banach–Saks
* Banach–Schauder (open mapping)
* Banach–Steinhaus (Uniform boundedness)
* Bessel's inequality
* Cauchy–Schwarz inequality
* Closed graph
* Closed range
* Eberlein–Šmulian
* Freudenthal spectral
* Gelfand–Mazur
* Gelfand–Naimark
* Goldstine
* Hahn–Banach (hyperplane separation)
* Kakutani fixed-point
* Krein–Milman
* Lomonosov's invariant subspace
* Mackey–Arens
* Mazur's lemma
* M. Riesz extension
* Riesz representation
* Parseval's identity
* Schauder fixed-point

Analysis|

* Abstract Wiener space
* Bochner space
* Differentiation in Fréchet spaces
* Derivatives (Fréchet
* Gateaux
* functional
* holomorphic)
* Integrals (Bochner
* Dunford
* Gelfand–Pettis
* regulated
* Paley–Wiener
* weak)
* Functional calculus (Borel
* continuous
* holomorphic)
* Inverse function theorem (Nash–Moser theorem)
* Measures (Lebesgue
* Projection-valued
* Vector)
* Weakly measurable function

Types of sets|

* Absolutely convex
* Absorbing
* Balanced
* Bounded
* Convex
* Convex cone (subset)
* Linear cone (subset)
* Radial
* Star-shaped
* Symmetric
* Zonotope

Subsets / set operations|

* Algebraic interior (core)
* Bounding points
* Convex hull
* Extreme point
* Interior
* Minkowski addition
* Polar

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