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 domain Ω. One might seek solutions σ ∈ 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/solutions σ ∈ 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 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 (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