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 modelled on Hilbert spaces.

Definition

Let X be a set. An atlas of class Cr, r ≥ 0, on X is a collection of pairs (called charts) (Uiφi), 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 i and j, φi(Ui ∩ Uj) is open in Ei;
3. the crossover map
$\varphi_{j} \circ \varphi_{i}^{-1} : \varphi_{i} (U_{i} \cap U_{j}) \to \varphi_{j} (U_{i} \cap U_{j})$
is an r-times continuously differentiable function for every i and j in I, i.e. the rth Fréchet derivative
$\mathrm{d}^{r} \big( \varphi_{j} \circ \varphi_{i}^{-1} \big) : \varphi_{i} (U_{i} \cap U_{j}) \to \mathrm{Lin} \big( E_{i}^{r}; E_{j} \big)$
exists and is a continuous function with respect to the Ei-norm topology on subsets of Ei and the operator norm topology on Lin(EirEj.)

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

$\varphi_{j} \circ \varphi_{i}^{-1} : \varphi_{i} (U_{i} \cap U_{j}) \to \varphi_{j} (U_{i} \cap U_{j})$

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 (Uiφi) 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 (Uφ) is called compatible with a given atlas { (Uiφi) | i ∈ I } if the crossover map

$\varphi_{i} \circ \varphi^{-1} : \varphi (U \cap U_{i}) \to \varphi_{i} (U \cap U_{i})$

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

• 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 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). 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, 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.
• Marsden, Ratiu, Abraham (2007). Manifolds, Tensor Analysis, and Applications. draft.