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## Summary

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

• 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 globally homeomorphic to $\mathbb {R} ^{n},$  or even an open subset of $\mathbb {R} ^{n}.$  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.