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.
is a collection of pairs (called charts)
such that One can then show that there is a unique topology on
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.
However, it is not a priori necessary that the Banach spaces
have a non-empty intersection, a quick examination of the derivative of the crossover map
must indeed be isomorphic as topological vector spaces.
isomorphic to a given Banach space
Hence, one can without loss of generality assume that, on each connected component of
is called compatible with a given atlas
-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
is then defined to be a choice of equivalence class of atlases on
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 a Banach manifold with an atlas containing a single, globally-defined chart (the identity map).
is an open subset of some Banach space then
It is by no means true that a finite-dimensional manifold of dimension
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
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.