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. Let be a set. An atlas of class on is a collection of pairs (called charts) such that each is a subset of and the union of the is the whole of ; each is a bijection from onto an open subset of some Banach space and for any indices is open in the crossover map is an -times continuously differentiable function for every that is, the th Fréchet derivative exists and is a continuous function with respect to the -norm topology on subsets of and the operator norm topology on One can then show that there is a unique topology on such that each is open and each 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 are equal to the same space the atlas is called an -atlas. However, it is not a priori necessary that the Banach spaces be the same space, or even isomorphic as topological vector spaces. However, if two charts and are such that and have a non-empty intersection, a quick examination of the derivative of the crossover map shows that and must indeed be isomorphic as topological vector spaces. Furthermore, the set of points for which there is a chart with in and isomorphic to a given Banach space is both open and closed. Hence, one can without loss of generality assume that, on each connected component of the atlas is an -atlas for some fixed A new chart is called compatible with a given atlas if the crossover map is an -times continuously differentiable function for every Two atlases are called compatible if every chart in one is compatible with the other atlas.