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

Let M be a Banach manifold of class C^p with p ≥ 0, called the base space; let E be a topological space, called the total space; let π : E → M be a surjective continuous map. Suppose that for each point x ∈ M, the fibre E_x = 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 X_i and a map τ_i
such that

the map τ_i is a homeomorphism commuting with the projection onto U_i, i.e. the following diagram commutes:
if U_i and U_j are two members of the open cover, then the map

The collection is called a trivialising covering for π : E → M, and the maps τ_i 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 X_i 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 fibre X. If M is a connected space then this is necessarily the case, since the set of points x ∈ 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 T_xV 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.
Similarly, the cotangent bundle T*M, whose fibre over a point x ∈ M is the Dual space#Continuous [dual space|topological dual space] to the tangent space at x:
There is a connection between Bochner spaces and Banach bundles. Consider, for example, the Bochner space X = 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 space H¹. 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.

Morphisms of Banach bundles

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 C^p, 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 :

the diagram
for each x₀ ∈ M there exist trivialising maps

Pull-back of a Banach bundle

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:

for each x ∈ M, _x = E_f;
there is a commutative diagram
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
if V is an open subset of N and U = f⁻¹, then