Banach manifold

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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.

[edit] Definition

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

each U_i is a subset of X and the union of the U_i is the whole of X;
each φ_i is a bijection from U_i onto an open subset φ_i(U_i) of some Banach space E_i, and for any i and j, φ_i(U_i ∩ U_j) is open in E_i;
the crossover map

$\varphi_{j} \circ \varphi_{i}^{-1} : \varphi_{i} (U_{i} \cap U_{j}) \to \varphi_{j} (U_{i} \cap U_{j})$

is an r-times continuously differentiable function for every i and j.

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 a priori 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

$\varphi_{j} \circ \varphi_{i}^{-1} : \varphi_{i} (U_{i} \cap U_{j}) \to \varphi_{j} (U_{i} \cap U_{j})$

shows that E_i and E_j must indeed be isomorphic as topological vector spaces. Furthermore, the set of points x ∈ X for which there is a chart (U_i, φ_i) 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, φ) is called compatible with a given atlas {(U_i, φ_i)| i ∈ I } if the crossover map

$\varphi_{i} \circ \varphi^{-1} : \varphi (U \cap U_{i}) \to \varphi_{i} (U \cap U_{i})$

is an r-times continuously differentiable function for every i. Two atlases are called compatible if every chart in one is compatible with the other atlas. Compatibility defines an equivalence relation on the set 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 modelled on E.