Banach manifold

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.

Definition

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

each $U_{i}$ is a subset of $X$ and the union of the $U_{i}$ is the whole of $X$ ;
each $\varphi _{i}$ is a bijection from $U_{i}$ onto an open 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 $E_{i};$
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 differentiable function for every $i,j\in I;$ that is, the $r$ th Fréchet 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 a continuous function with respect to the $E_{i}$ -norm topology on subsets of $E_{i}$ and the operator norm topology on $\operatorname {Lin} \left(E_{i}^{r};E_{j}\right).$

One can then show that there is a unique topology on $X$ such that each $U_{i}$ is open and each $\varphi _{i}$ 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 $E_{i}$ 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 $E_{i}$ be the same space, or even isomorphic as topological 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-empty intersection, a quick examination of the derivative of 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 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 $(U,\varphi )$ is called compatible with 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 an equivalence relation on the class of all possible atlases on $X.$

A $C^{r}$ -manifold structure 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 $X$ 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 $(X,\|\,\cdot \,\|)$ 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 $\mathbb {R} ^{n},$ 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 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 $\ell ^{2}$ ). 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.

References

^ 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" (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.

[FOOTNOTEHenderson1969-1] Henderson 1969.

[1]

v t e Analysis in topological vector spaces
Basic concepts	Abstract Wiener space Classical Wiener space Bochner space Convex series Cylinder set measure Infinite-dimensional vector function Matrix calculus Vector calculus
Derivatives	Differentiable vector–valued functions from Euclidean space Differentiation in Fréchet spaces Fréchet derivative Total Functional derivative Gateaux derivative Directional Generalizations of the derivative Hadamard derivative Holomorphic Quasi-derivative
Measurability	Besov measure Cylinder set measure Canonical Gaussian Classical Wiener measure Measure like set functions infinite-dimensional Gaussian measure Projection-valued Vector Bochner / Weakly / Strongly measurable function Radonifying function
Integrals	Bochner Direct integral Dunford Gelfand–Pettis/Weak Regulated Paley–Wiener
Results	Cameron–Martin theorem Inverse function theorem Nash–Moser theorem Feldman–Hájek theorem No infinite-dimensional Lebesgue measure Sazonov's theorem Structure theorem for Gaussian measures
Related	Crinkled arc Covariance operator
Functional calculus	Borel functional calculus Continuous functional calculus Holomorphic functional calculus
Applications	Banach manifold (bundle) Convenient vector space Choquet theory Fréchet manifold Hilbert manifold

v t e Banach space topics
Types of Banach spaces	Asplund Banach list Banach lattice Grothendieck Hilbert Inner product space Polarization identity (Polynomially) Reflexive Riesz L-semi-inner product (B Strictly Uniformly) convex Uniformly smooth (Injective Projective) Tensor product (of Hilbert spaces)
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Function space Topologies	Banach–Mazur compactum Dual Dual space Dual norm Operator Ultraweak Weak polar operator Strong polar operator Ultrastrong Uniform convergence
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Operator theory	Banach algebras C-algebras Operator space Spectrum C-algebra radius Spectral theory of ODEs Spectral theorem Polar decomposition Singular value decomposition
Theorems	Anderson–Kadec 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 Parseval's identity Riesz's lemma Riesz representation Robinson-Ursescu Schauder fixed-point
Analysis	Abstract Wiener space Banach manifold bundle Bochner space Convex series Differentiation in Fréchet spaces Derivatives Fréchet Gateaux functional holomorphic quasi Integrals Bochner Dunford Gelfand–Pettis regulated Paley–Wiener weak Functional calculus Borel continuous holomorphic Measures Lebesgue Projection-valued Vector Weakly / Strongly measurable function
Types of sets	Absolutely convex Absorbing Affine Balanced/Circled Bounded Convex Convex cone (subset) Convex series related ((cs, lcs)-closed, (cs, bcs)-complete, (lower) ideally convex, (Hx), and (Hwx)) Linear cone (subset) Radial Radially convex/Star-shaped Symmetric Zonotope
Subsets / set operations	Affine hull (Relative) Algebraic interior (core) Bounding points Convex hull Extreme point Interior Linear span Minkowski addition Polar (Quasi) Relative interior
Examples	Absolute continuity AC $ba(\Sigma )$ c space Banach coordinate BK Besov $B_{p,q}^{s}(\mathbb {R} )$ Birnbaum–Orlicz Bounded variation BV Bs space Continuous C(K) with K compact Hausdorff Hardy H^p Hilbert H Morrey–Campanato $L^{\lambda ,p}(\Omega )$ ℓ^p $\ell ^{\infty }$ L^p $L^{\infty }$ weighted Schwartz $S\left(\mathbb {R} ^{n}\right)$ Segal–Bargmann F Sequence space Sobolev W^k,p Sobolev inequality Triebel–Lizorkin Wiener amalgam $W(X,L^{p})$
Applications	Differential operator Finite element method Mathematical formulation of quantum mechanics Ordinary Differential Equations (ODEs) Validated numerics

Definition

Examples

Classification up to homeomorphism

See also

References