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 [math]\displaystyle{ X }[/math] be a set. An atlas of class [math]\displaystyle{ C^r, }[/math] [math]\displaystyle{ r \geq 0, }[/math] on [math]\displaystyle{ X }[/math] is a collection of pairs (called charts) [math]\displaystyle{ \left(U_i, \varphi_i\right), }[/math] [math]\displaystyle{ i \in I, }[/math] such that

1. each [math]\displaystyle{ U_i }[/math] is a subset of [math]\displaystyle{ X }[/math] and the union of the [math]\displaystyle{ U_i }[/math] is the whole of [math]\displaystyle{ X }[/math];
2. each [math]\displaystyle{ \varphi_i }[/math] is a bijection from [math]\displaystyle{ U_i }[/math] onto an open subset [math]\displaystyle{ \varphi_i\left(U_i\right) }[/math] of some Banach space [math]\displaystyle{ E_i, }[/math] and for any indices [math]\displaystyle{ i \text{ and } j, }[/math] [math]\displaystyle{ \varphi_i\left(U_i \cap U_j\right) }[/math] is open in [math]\displaystyle{ E_i; }[/math]
3. the crossover map [math]\displaystyle{ \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) }[/math] is an [math]\displaystyle{ r }[/math]-times continuously differentiable function for every [math]\displaystyle{ i, j \in I; }[/math] that is, the [math]\displaystyle{ r }[/math]th Fréchet derivative [math]\displaystyle{ \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) }[/math] exists and is a continuous function with respect to the [math]\displaystyle{ E_i }[/math]-norm topology on subsets of [math]\displaystyle{ E_i }[/math] and the operator norm topology on [math]\displaystyle{ \operatorname{Lin}\left(E_i^r; E_j\right). }[/math]

One can then show that there is a unique topology on [math]\displaystyle{ X }[/math] such that each [math]\displaystyle{ U_i }[/math] is open and each [math]\displaystyle{ \varphi_i }[/math] 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 [math]\displaystyle{ E_i }[/math] are equal to the same space [math]\displaystyle{ E, }[/math] the atlas is called an [math]\displaystyle{ E }[/math]-atlas. However, it is not a priori necessary that the Banach spaces [math]\displaystyle{ E_i }[/math] be the same space, or even isomorphic as topological vector spaces. However, if two charts [math]\displaystyle{ \left(U_i, \varphi_i\right) }[/math] and [math]\displaystyle{ \left(U_j, \varphi_j\right) }[/math] are such that [math]\displaystyle{ U_i }[/math] and [math]\displaystyle{ U_j }[/math] have a non-empty intersection, a quick examination of the derivative of the crossover map [math]\displaystyle{ \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) }[/math] shows that [math]\displaystyle{ E_i }[/math] and [math]\displaystyle{ E_j }[/math] must indeed be isomorphic as topological vector spaces. Furthermore, the set of points [math]\displaystyle{ x \in X }[/math] for which there is a chart [math]\displaystyle{ \left(U_i, \varphi_i\right) }[/math] with [math]\displaystyle{ x }[/math] in [math]\displaystyle{ U_i }[/math] and [math]\displaystyle{ E_i }[/math] isomorphic to a given Banach space [math]\displaystyle{ E }[/math] is both open and closed. Hence, one can without loss of generality assume that, on each connected component of [math]\displaystyle{ X, }[/math] the atlas is an [math]\displaystyle{ E }[/math]-atlas for some fixed [math]\displaystyle{ E. }[/math]

A new chart [math]\displaystyle{ (U, \varphi) }[/math] is called compatible with a given atlas [math]\displaystyle{ \left\{\left(U_i, \varphi_i\right) : i \in I\right\} }[/math] if the crossover map [math]\displaystyle{ \varphi_i \circ \varphi^{-1} : \varphi\left(U \cap U_i\right) \to \varphi_i\left(U \cap U_i\right) }[/math] is an [math]\displaystyle{ r }[/math]-times continuously differentiable function for every [math]\displaystyle{ i \in I. }[/math] 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 [math]\displaystyle{ X. }[/math]

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

Examples

Every Banach space can be canonically identified as a Banach manifold. If [math]\displaystyle{ (X, \|\,\cdot\,\|) }[/math] is a Banach space, then [math]\displaystyle{ X }[/math] is a Banach manifold with an atlas containing a single, globally-defined chart (the identity map).

Similarly, if [math]\displaystyle{ U }[/math] is an open subset of some Banach space then [math]\displaystyle{ U }[/math] 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 [math]\displaystyle{ n }[/math] is globally homeomorphic to [math]\displaystyle{ \Reals^n, }[/math] or even an open subset of [math]\displaystyle{ \Reals^n. }[/math] 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 [math]\displaystyle{ X }[/math] can be embedded as an open subset of the infinite-dimensional, separable Hilbert space, [math]\displaystyle{ H }[/math] (up to linear isomorphism, there is only one such space, usually identified with [math]\displaystyle{ \ell^2 }[/math]). 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 [math]\displaystyle{ X. }[/math] 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
• Fréchet manifold
• 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. 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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