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 modelled on Hilbert spaces.

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

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 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

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