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) i \in I, such that 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 and are such that U_i and U_j have a non-empty intersection, a quick examination of the derivative of the crossover map 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 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 is called compatible with a given atlas if the crossover map 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 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 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 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.