Banach bundle
In mathematics, a Banach bundle is a vector bundle each of whose fibres is a Banach space, i.e. a complete normed vector space, possibly of infinite dimension.
Definition of a Banach bundle[edit]
Let M be a Banach manifold of class Cp with p ≥ 0, called the base space; let E be a topological space, called the total space; let
be an open cover of M. Suppose also that for each i ∈ I, there is a Banach space Xi and a map
such that
- the map
τ i is a homeomorphism commuting with the projection onto Ui, i.e. the following diagram commutes:
- and for each x ∈ Ui the induced map
τ ix on the fibre Ex
- is an invertible continuous linear map, i.e. an isomorphism in the category of topological vector spaces;
- if Ui and Uj are two members of the open cover, then the map
- is a morphism (a differentiable map of class Cp), where Lin(X; Y) denotes the space of all continuous linear maps from a topological vector space X to another topological vector space Y.
The collection {(Ui,
If all the spaces Xi are isomorphic as topological vector spaces, then they can be assumed all to be equal to the same space X. In this case,
for a given space X is both open and closed.
In the finite-dimensional case, the second condition above is implied by the first.
Examples of Banach bundles[edit]
- If V is any Banach space, the tangent space TxV to V at any point x ∈ V is isomorphic in an obvious way to V itself. The tangent bundle TV of V is then a Banach bundle with the usual projection
- This bundle is "trivial" in the sense that TV admits a globally defined trivialising map: the identity function
- If M is any Banach manifold, the tangent bundle TM of M forms a Banach bundle with respect to the usual projection, but it may not be trivial.
- Similarly, the cotangent bundle T*M, whose fibre over a point x ∈ M is the topological dual space to the tangent space at x:
- also forms a Banach bundle with respect to the usual projection onto M.
- There is a connection between Bochner spaces and Banach bundles. Consider, for example, the Bochner space X = L²([0, T]; H1(
Ω )), which might arise as a useful object when studying the heat equation on a domainΩ . One might seek solutionsσ ∈ X to the heat equation; for each time t,σ (t) is a function in the Sobolev space H1(Ω ). One could also think of Y = [0, T] × H1(Ω ), which as a Cartesian product also has the structure of a Banach bundle over the manifold [0, T] with fibre H1(Ω ), in which case elements/solutionsσ ∈ X are cross sections of the bundle Y of some specified regularity (L², in fact). If the differential geometry of the problem in question is particularly relevant, the Banach bundle point of view might be advantageous.
Morphisms of Banach bundles[edit]
The collection of all Banach bundles can be made into a category by defining appropriate morphisms.
Let
For f to be a morphism means simply that f is a continuous map of topological spaces. If the manifolds M and M′ are both of class Cp, then the requirement that f0 be a morphism is the requirement that it be a p-times continuously differentiable function. These two morphisms are required to satisfy two conditions (again, the second one is redundant in the finite-dimensional case):
- the diagram
- commutes, and, for each x ∈ M, the induced map
- is a continuous linear map;
- for each x0 ∈ M there exist trivialising maps
- such that x0 ∈ U, f0(x0) ∈ U′,
- and the map
- is a morphism (a differentiable map of class Cp).
Pull-back of a Banach bundle[edit]
One can take a Banach bundle over one manifold and use the pull-back construction to define a new Banach bundle on a second manifold.
Specifically, let
- for each x ∈ M, (f*E)x = Ef(x);
- there is a commutative diagram
- with the top horizontal map being the identity on each fibre;
- if E is trivial, i.e. equal to N × X for some Banach space X, then f*E is also trivial and equal to M × X, and
- is the projection onto the first coordinate;
- if V is an open subset of N and U = f−1(V), then
- and there is a commutative diagram
- where the maps at the "front" and "back" are the same as those in the previous diagram, and the maps from "back" to "front" are (induced by) the inclusions.
References[edit]
- Lang, Serge (1972). Differential manifolds. Reading, Mass.–London–Don Mills, Ont.: Addison-Wesley Publishing Co., Inc.