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Banach manifold

## 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 ${\displaystyle X}$  be a set. An atlas of class ${\displaystyle C^{r},}$  ${\displaystyle r\geq 0,}$  on ${\displaystyle X}$  is a collection of pairs (called charts) ${\displaystyle \left(U_{i},\varphi _{i}\right),}$  ${\displaystyle i\in I,}$  such that

1. each ${\displaystyle U_{i}}$  is a subset of ${\displaystyle X}$  and the union of the ${\displaystyle U_{i}}$  is the whole of ${\displaystyle X}$ ;
2. each ${\displaystyle \varphi _{i}}$  is a bijection from ${\displaystyle U_{i}}$  onto an open subset ${\displaystyle \varphi _{i}\left(U_{i}\right)}$  of some Banach space ${\displaystyle E_{i},}$  and for any indices ${\displaystyle i{\text{ and }}j,}$  ${\displaystyle \varphi _{i}\left(U_{i}\cap U_{j}\right)}$  is open in ${\displaystyle E_{i};}$
3. the crossover map
${\displaystyle \varphi _{j}\circ \varphi _{i}^{-1}:\varphi _{i}\left(U_{i}\cap U_{j}\right)\to \varphi _{j}\left(U_{i}\cap U_{j}\right)}$

is an ${\displaystyle r}$ -times continuously differentiable function for every ${\displaystyle i,j\in I;}$  that is, the ${\displaystyle r}$ th Fréchet derivative
${\displaystyle \mathrm {d} ^{r}\left(\varphi _{j}\circ \varphi _{i}^{-1}\right):\varphi _{i}\left(U_{i}\cap U_{j}\right)\to \mathrm {Lin} \left(E_{i}^{r};E_{j}\right)}$

exists and is a continuous function with respect to the ${\displaystyle E_{i}}$ -norm topology on subsets of ${\displaystyle E_{i}}$  and the operator norm topology on ${\displaystyle \operatorname {Lin} \left(E_{i}^{r};E_{j}\right).}$

One can then show that there is a unique topology on ${\displaystyle X}$  such that each ${\displaystyle U_{i}}$  is open and each ${\displaystyle \varphi _{i}}$  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 ${\displaystyle E_{i}}$  are equal to the same space ${\displaystyle E,}$  the atlas is called an ${\displaystyle E}$ -atlas. However, it is not a priori necessary that the Banach spaces ${\displaystyle E_{i}}$  be the same space, or even isomorphic as topological vector spaces. However, if two charts ${\displaystyle \left(U_{i},\varphi _{i}\right)}$  and ${\displaystyle \left(U_{j},\varphi _{j}\right)}$  are such that ${\displaystyle U_{i}}$  and ${\displaystyle U_{j}}$  have a non-empty intersection, a quick examination of the derivative of the crossover map

${\displaystyle \varphi _{j}\circ \varphi _{i}^{-1}:\varphi _{i}\left(U_{i}\cap U_{j}\right)\to \varphi _{j}\left(U_{i}\cap U_{j}\right)}$

shows that ${\displaystyle E_{i}}$  and ${\displaystyle E_{j}}$  must indeed be isomorphic as topological vector spaces. Furthermore, the set of points ${\displaystyle x\in X}$  for which there is a chart ${\displaystyle \left(U_{i},\varphi _{i}\right)}$  with ${\displaystyle x}$  in ${\displaystyle U_{i}}$  and ${\displaystyle E_{i}}$  isomorphic to a given Banach space ${\displaystyle E}$  is both open and closed. Hence, one can without loss of generality assume that, on each connected component of ${\displaystyle X,}$  the atlas is an ${\displaystyle E}$ -atlas for some fixed ${\displaystyle E.}$

A new chart ${\displaystyle (U,\varphi )}$  is called compatible with a given atlas ${\displaystyle \left\{\left(U_{i},\varphi _{i}\right):i\in I\right\}}$  if the crossover map

${\displaystyle \varphi _{i}\circ \varphi ^{-1}:\varphi \left(U\cap U_{i}\right)\to \varphi _{i}\left(U\cap U_{i}\right)}$

is an ${\displaystyle r}$ -times continuously differentiable function for every ${\displaystyle i\in I.}$  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 ${\displaystyle X.}$

A ${\displaystyle C^{r}}$ -manifold structure on ${\displaystyle X}$  is then defined to be a choice of equivalence class of atlases on ${\displaystyle X}$  of class ${\displaystyle C^{r}.}$  If all the Banach spaces ${\displaystyle E_{i}}$  are isomorphic as topological vector spaces (which is guaranteed to be the case if ${\displaystyle X}$  is connected), then an equivalent atlas can be found for which they are all equal to some Banach space ${\displaystyle E.}$  ${\displaystyle X}$  is then called an ${\displaystyle E}$ -manifold, or one says that ${\displaystyle X}$  is modeled on ${\displaystyle E.}$

## Examples

• If ${\displaystyle (X,\|\,\cdot \,\|)}$  is a Banach space, then ${\displaystyle X}$  is a Banach manifold with an atlas containing a single, globally-defined chart (the identity map).
• Similarly, if ${\displaystyle U}$  is an open subset of some Banach space then ${\displaystyle U}$  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 ${\displaystyle n}$  is globally homeomorphic to ${\displaystyle \mathbb {R} ^{n},}$  or even an open subset of ${\displaystyle \mathbb {R} ^{n}.}$  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 ${\displaystyle X}$  can be embedded as an open subset of the infinite-dimensional, separable Hilbert space, ${\displaystyle H}$  (up to linear isomorphism, there is only one such space, usually identified with ${\displaystyle \ell ^{2}}$ ). 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 ${\displaystyle X.}$  Thus, in the infinite-dimensional, separable, metric case, the "only" Banach manifolds are the open subsets of Hilbert space.