# Kan extension

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 modelled on Hilbert spaces.

## Definition

Let *X* be a set. An **atlas of class** *C*^{r}, *r* ≥ 0, on *X* is a collection of pairs (called **charts**) (*U*_{i}, *φ*_{i}), *i* ∈ *I*, such that

- each
*U*_{i}is a subset of*X*and the union of the*U*_{i}is the whole of*X*; - each
*φ*_{i}is a bijection from*U*_{i}onto an open subset*φ*_{i}(*U*_{i}) of some Banach space*E*_{i}, and for any*i*and*j*,*φ*_{i}(*U*_{i}∩*U*_{j}) is open in*E*_{i}; - the crossover map

- is an
*r*-times continuously differentiable function for every*i*and*j*in*I*, i.e. the*r*^{th}Fréchet derivative

- 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 Lin(*E*_{i}^{r};*E*_{j}.)

One can then show that there is a unique topology on *X* such that each *U*_{i} is open and each *φ*_{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 (*U*_{i}, *φ*_{i}) and (*U*_{j}, *φ*_{j}) are such that *U*_{i} and *U*_{j} have a non-empty intersection, a quick examination of the derivative of the crossover map

shows that *E*_{i} and *E*_{j} must indeed be isomorphic as topological vector spaces. Furthermore, the set of points *x* ∈ *X* for which there is a chart (*U*_{i}, *φ*_{i}) 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*, *φ*) is called **compatible** with a given atlas { (*U*_{i}, *φ*_{i}) | *i* ∈ *I* } if the crossover map

is an *r*-times continuously differentiable function for every *i* ∈ *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*, ||⋅||) 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 **R**^{n}, or even an open subset of **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). 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.

## References

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