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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.