Kan extension

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 C^r, r ≥ 0, on X is a collection of pairs (called charts) (U_i, φ_i), i ∈ I, such that

1. each U_i is a subset of X and the union of the U_i is the whole of X;
2. each φ_i is a bijection from U_i onto an open subset φ_i(U_i) of some Banach space E_i, and for any i and j, φ_i(U_i ∩ U_j) is open in E_i;
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 r^th 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 E_i-norm topology on subsets of E_i and the operator norm topology on Lin(E_i^r; E_j.)

One can then show that there is a unique topology on X such that each U_i is open and each φ_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 (U_i, φ_i) and (U_j, φ_j) are such that U_i and U_j have a non-empty intersection, a quick examination of the derivative of the crossover map

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

shows that E_i and E_j must indeed be isomorphic as topological vector spaces. Furthermore, the set of points x ∈ X for which there is a chart (U_i, φ_i) 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, φ) is called compatible with a given atlas { (U_i, φ_i) | 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 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

• 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

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