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 $$X$$ be a set. An atlas of class $$C^r,$$ $$r \geq 0,$$ on $$X$$ is a collection of pairs (called charts) $$\left(U_i, \varphi_i\right),$$ $$i \in I,$$ such that


 * 1) each $$U_i$$ is a subset of $$X$$ and the union of the $$U_i$$ is the whole of $$X$$;
 * 2) each $$\varphi_i$$ is a bijection from $$U_i$$ onto an open subset $$\varphi_i\left(U_i\right)$$ of some Banach space $$E_i,$$ and for any indices $$i \text{ and } j,$$ $$\varphi_i\left(U_i \cap U_j\right)$$ is open in $$E_i;$$
 * 3) the crossover map $$\varphi_j \circ \varphi_i^{-1} : \varphi_i\left(U_i \cap U_j\right) \to \varphi_j\left(U_i \cap U_j\right)$$ is an $r$-times continuously differentiable function for every $$i, j \in I;$$ that is, the $$r$$th Fréchet derivative $$\mathrm{d}^r\left(\varphi_j \circ \varphi_i^{-1}\right) : \varphi_i\left(U_i \cap U_j\right) \to \mathrm{Lin}\left(E_i^r; E_j\right)$$ exists and is a continuous function with respect to the $$E_i$$-norm topology on subsets of $$E_i$$ and the operator norm topology on $$\operatorname{Lin}\left(E_i^r; E_j\right).$$

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 $$\left(U_i, \varphi_i\right)$$ and $$\left(U_j, \varphi_j\right)$$ are such that $$U_i$$ and $$U_j$$ have a non-empty intersection, a quick examination of the derivative of the crossover map $$\varphi_j \circ \varphi_i^{-1} : \varphi_i\left(U_i \cap U_j\right) \to \varphi_j\left(U_i \cap U_j\right)$$ 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 $$\left(U_i, \varphi_i\right)$$ 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 $$(U, \varphi)$$ is called compatible with a given atlas $$\left\{\left(U_i, \varphi_i\right) : i \in I\right\}$$ if the crossover map $$\varphi_i \circ \varphi^{-1} : \varphi\left(U \cap U_i\right) \to \varphi_i\left(U \cap U_i\right)$$ 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 $$(X, \|\,\cdot\,\|)$$ 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.