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Banach manifold - Wikipedia

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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 modeled on Hilbert spaces.

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.}

Every Banach space can be canonically identified as a Banach manifold. 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

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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[1] 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.