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 formula_1 be a set . An atlas of class formula_2 formula_3 on formula_1 is a collection of pairs (called charts) formula_5 formula_6 such that One can then show that there is a unique topology on formula_1 such that each formula_8 is open and each formula_9 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 formula_10 are equal to the same space formula_11 the atlas is called an formula_12-atlas. However, it is not "" necessary that the Banach spaces formula_10 be  the same space, or even isomorphic as topological vector spaces. However, if two charts formula_14 and formula_15 are such that formula_8 and formula_17 have a non-empty intersection, a quick examination of the derivative of the crossover map formula_18 shows that formula_10 and formula_20 must indeed be isomorphic as topological vector spaces. Furthermore, the set of points formula_21 for which there is a chart formula_14 with formula_23 in formula_8 and formula_10 isomorphic to a given Banach space formula_12 is both open and closed. Hence, one can without loss of generality assume that, on each connected component of formula_27 the atlas  is an formula_12-atlas for some fixed formula_29 A new chart formula_30 is called compatible with a given atlas formula_31 if the crossover map formula_32 is an formula_33-times continuously differentiable function for every formula_34 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 formula_35 A formula_36-manifold structure on formula_1 is then defined to be a choice of equivalence class of atlases on formula_1 of class formula_39 If all the Banach spaces formula_10 are isomorphic as topological vector spaces  (which is guaranteed to be the case if formula_1 is connected), then an equivalent atlas can be found for which they are all equal to some Banach space formula_29 formula_1 is then called an formula_12-manifold, or one says that formula_1 is modeled on formula_29 Classification up to homeomorphism. It is by no means true that a finite-dimensional manifold of dimension formula_51 is homeomorphic to formula_52 or even an open subset of formula_53 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 formula_1 can be embedded as an open subset of the infinite-dimensional, separable Hilbert space, formula_55 (up to linear isomorphism, there is only one such space, usually identified with formula_56). 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 formula_35 Thus, in the infinite-dimensional, separable, metric case, the "only" Banach manifolds are the open subsets of Hilbert space.