Manifold modeled on Banach spaces
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
be a set. An atlas of class
on
is a collection of pairs (called charts)
such that
- each
is a subset of
and the union of the
is the whole of
;
- each
is a bijection from
onto an open subset
of some Banach space
and for any indices
is open in 
- the crossover map

is an
-times continuously differentiable function for every
that is, the
th Fréchet derivative 
exists and is a continuous function with respect to the
-norm topology on subsets of
and the operator norm topology on 
One can then show that there is a unique topology on
such that each
is open and each
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
are equal to the same space
the atlas is called an
-atlas. However, it is not a priori necessary that the Banach spaces
be the same space, or even isomorphic as topological vector spaces. However, if two charts
and
are such that
and
have a non-empty intersection, a quick examination of the derivative of the crossover map

shows that
and
must indeed be isomorphic as topological vector spaces. Furthermore, the set of points
for which there is a chart
with
in
and
isomorphic to a given Banach space
is both open and closed. Hence, one can without loss of generality assume that, on each connected component of
the atlas is an
-atlas for some fixed
A new chart
is called compatible with a given atlas
if the crossover map

is an
-times continuously differentiable function for every
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
A
-manifold structure on
is then defined to be a choice of equivalence class of atlases on
of class
If all the Banach spaces
are isomorphic as topological vector spaces (which is guaranteed to be the case if
is connected), then an equivalent atlas can be found for which they are all equal to some Banach space
is then called an
-manifold, or one says that
is modeled on
Classification up to homeomorphism
It is by no means true that a finite-dimensional manifold of dimension
is globally homeomorphic to
or even an open subset of
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
can be embedded as an open subset of the infinite-dimensional, separable Hilbert space,
(up to linear isomorphism, there is only one such space, usually identified with
). 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
Thus, in the infinite-dimensional, separable, metric case, the "only" Banach manifolds are the open subsets of Hilbert space.