Inmathematics, aBanach bundle is avector bundle each of whose fibres is aBanach space, i.e. acompletenormed vector space, possibly of infinite dimension.

LetM be aBanach manifold of classC^p withp ≥ 0, called thebase space; letE be atopological space, called thetotal space; letπ :E →M be asurjectivecontinuous map. Suppose that for each pointx ∈M, thefibreE_x =π⁻¹(x) has been given the structure of a Banach space. Let

${\displaystyle \{U_i|i\in I\}}$

be anopen cover ofM. Suppose also that for eachi ∈I, there is a Banach spaceX_i and a mapτ_i

${\displaystyle {\tau }_i:{\pi }^{-1}(U_i)\to U_i\times X_i}$

such that

• the mapτ_i is ahomeomorphism commuting with the projection ontoU_i, i.e. the followingdiagram commutes:

and for eachx ∈U_i the induced mapτ_ix on the fibreE_x

${\displaystyle {\tau }_{ix}:{\pi }^{-1}(x)\to X_i}$

is aninvertiblecontinuous linear map, i.e. anisomorphism in thecategory oftopological vector spaces;

• ifU_i andU_j are two members of the open cover, then the map

${\displaystyle U_i\cap U_j\to \mathrm{L}\mathrm{i}\mathrm{n}(X_i;X_j)}$

${\displaystyle x\mapsto ({\tau }_j\circ {\tau }_i^{-1}{)}_x}$

is amorphism (a differentiable map of classC^p), where Lin(X;Y) denotes the space of all continuous linear maps from a topological vector spaceX to another topological vector spaceY.

The collection {(U_i,τ_i)|i∈I} is called atrivialising covering forπ :E →M, and the mapsτ_i are calledtrivialising maps. Two trivialising coverings are said to beequivalent if their union again satisfies the two conditions above. Anequivalence class of such trivialising coverings is said to determine the structure of aBanach bundle onπ :E →M.

If all the spacesX_i are isomorphic as topological vector spaces, then they can be assumed all to be equal to the same spaceX. In this case,π :E →M is said to be aBanach bundle with fibreX. IfM is aconnected space then this is necessarily the case, since the set of pointsx ∈M for which there is a trivialising map

${\displaystyle {\tau }_{ix}:{\pi }^{-1}(x)\to X}$

for a given spaceX is bothopen andclosed.

In the finite-dimensional case, the second condition above is implied by the first.

• IfV is any Banach space, thetangent space T_xV toV at any pointx ∈V is isomorphic in an obvious way toV itself. Thetangent bundle TV ofV is then a Banach bundle with the usual projection

${\displaystyle \pi :\mathrm{T}V\to V;}$

${\displaystyle (x,v)\mapsto x.}$

This bundle is "trivial" in the sense that TV admits a globally defined trivialising map: theidentity function

${\displaystyle \tau =\mathrm{i}\mathrm{d}:{\pi }^{-1}(V)=\mathrm{T}V\to V\times V;}$

${\displaystyle (x,v)\mapsto (x,v).}$

• IfM is any Banach manifold, the tangent bundle TM ofM forms a Banach bundle with respect to the usual projection, but it may not be trivial.
• Similarly, thecotangent bundle T*M, whose fibre over a pointx ∈M is thetopological dual space to the tangent space atx:

${\displaystyle {\pi }^{-1}(x)={\mathrm{T}}_x^{\ast }M=({\mathrm{T}}_{x}M{)}^{\ast };}$

also forms a Banach bundle with respect to the usual projection ontoM.

• There is a connection betweenBochner spaces and Banach bundles. Consider, for example, the Bochner spaceX = L²([0, T]; H¹(Ω)), which might arise as a useful object when studying theheat equation on a domain Ω. One might seek solutionsσ ∈ X to the heat equation; for each timet,σ(t) is a function in theSobolev spaceH¹(Ω). One could also think ofY = [0, T] × H¹(Ω), which as aCartesian product also has the structure of a Banach bundle over the manifold [0, T] with fibreH¹(Ω), in which case elements/solutionsσ ∈ X arecross sections of the bundleY 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.

The collection of all Banach bundles can be made into a category by defining appropriate morphisms.

Letπ :E →M andπ′ :E′ →M′ be two Banach bundles. ABanach bundle morphism from the first bundle to the second consists of a pair of morphisms

${\displaystyle f_0:M\to M^{\prime };}$

${\displaystyle f:E\to E^{\prime }.}$

Forf to be a morphism means simply thatf is a continuous map of topological spaces. If the manifoldsM andM′ are both of classC^p, then the requirement thatf₀ be a morphism is the requirement that it be ap-times continuouslydifferentiable 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 eachx ∈M, the induced map

${\displaystyle f_x:E_x\to E_{f_0(x)}^{\prime }}$

is a continuous linear map;

• for eachx₀ ∈M there exist trivialising maps

${\displaystyle \tau :{\pi }^{-1}(U)\to U\times X}$

${\displaystyle {\tau }^{\prime }:{\pi }^{\prime -1}(U^{\prime })\to U^{\prime }\times X^{\prime }}$

such thatx₀ ∈U,f₀(x₀) ∈U′,

${\displaystyle f_0(U)\subseteq U^{\prime }}$

and the map

${\displaystyle U\to \mathrm{L}\mathrm{i}\mathrm{n}(X;X^{\prime })}$

${\displaystyle x\mapsto {\tau }_{f_0(x)}^{\prime }\circ f_x\circ {\tau }^{-1}}$

is a morphism (a differentiable map of classC^p).

One can take a Banach bundle over one manifold and use thepull-back construction to define a new Banach bundle on a second manifold.

Specifically, letπ :E →N be a Banach bundle andf :M →N a differentiable map (as usual, everything isC^p). Then thepull-back ofπ :E →N is the Banach bundlef*π :f*E →M satisfying the following properties:

• for eachx ∈M, (f*E)_x =E_f(x);
• there is a commutative diagram

with the top horizontal map being the identity on each fibre;

• ifE is trivial, i.e. equal toN ×X for some Banach spaceX, thenf*E is also trivial and equal toM ×X, and

${\displaystyle f^{\ast }\pi :f^{\ast }E=M\times X\to M}$

is the projection onto the first coordinate;

• ifV is an open subset ofN andU =f⁻¹(V), then

${\displaystyle f^{\ast }(E_V)=(f^{\ast }E{)}_U}$

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.

• Lang, Serge (1972).Differential manifolds. Reading, Mass.–London–Don Mills, Ont.: Addison-Wesley Publishing Co., Inc.

• Banach spaces
• Differential geometry
• Generalized manifolds
• Manifolds
• Nonlinear functional analysis
• Structures on manifolds
• Vector bundles