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

  1. each U_i is a subset of X and the union of the U_i is the whole of X;
  2. 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;
  3. 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 rth 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 \Reals^n, or even an open subset of \Reals^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.

See also

  • Global analysis – which uses Banach manifolds and other kinds of infinite-dimensional manifolds

References

  1. ^ Henderson 1969.