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 Cp 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 Ex = π−1(x) 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: and for each x ∈ Ui the induced map τix 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, τi)|i∈I} 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 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 TxV to V at any point x ∈ V is isomorphic in an obvious way to V itself. The