Let M be a Banach manifold of classCp with p ≥ 0, called the base space; let E be a topological space, called the total space; let π : E → M be a surjectivecontinuous map. Suppose that for each point x ∈ M, the fibreEx = π−1 has been given the structure of a Banach space. Let be an open cover of M. Suppose also that for each i ∈ I, there is a Banach space Xi and a map τi such that
the map τi is a homeomorphism commuting with the projection onto Ui, i.e. the following diagram commutes:
if Ui and Uj are two members of the open cover, then the map
The collection is called a trivialising covering for π : E → M, and the maps τi are called trivialising maps. Two trivialising coverings are said to be equivalent if their union again satisfies the two conditions above. An equivalence class of such trivialising coverings is said to determine the structure of a Banach bundle on π : E → M. 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, π : E → M is said to be a Banach bundle with fibreX. If M is a connected space then this is necessarily the case, since the set of pointsx ∈ M for which there is a trivialising map 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
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
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.
There is a connection between Bochner spaces and Banach bundles. Consider, for example, the Bochner spaceX = L², which might arise as a useful object when studying the heat equation on a domain Ω. One might seek solutions σ ∈ X to the heat equation; for each time t, σ is a function in the Sobolev spaceH1. One could also think of Y = × H1, which as a Cartesian product also has the structure of a Banach bundle over the manifold with fibre H1, in which case elements/solutions σ ∈ X are cross sections of the bundle Y of some specified regularity. If the differential geometry of the problem in question is particularly relevant, the Banach bundle point of view might be advantageous.
The collection of all Banach bundles can be made into a category by defining appropriate morphisms. Let π : E → M and π′ : E′ → M′ be two Banach bundles. A Banach bundle morphism from the first bundle to the second consists of a pair of morphisms 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 :
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 π : E → N be a Banach bundle and f : M → N a differentiable map. Then the pull-back of π : E → N is the Banach bundle f*π : f*E → M satisfying the following properties:
