A Banach bundle is an open (necessarily surjective) continuous map of Hausdorff topological spaces , each of whose fibers carries a structure of a complex Banach space, this structure being continuous in the base point (in other words, the global operations of multiplication by a scalar, of addition and of taking the norm are continuous) and such that for every net , if and , then .
We distinguish a different concept of Banach algebraic bundle, where the base space is also a Banach algebra and the multiplication is defined as a map (not only ), that is we can multiply the points in different fibers, and .
A Banach bundle is a Hilbert bundle if each fiber is a separable Hilbert space. As usual, the inner product can be obtained by the polarization formula from the norm of a Banach space if the norm satisfies the parallelogram identity. From this, we infer that for Hilbert bundles, the inner product is continuous as a map . Hilbert bundles are important in the study of induced representations of locally compact groups, and Mackey theory? in particular; more recently their study is connected to the study of Hilbert modules.
A morphism of Banach bundles over the same base is a morphism of total spaces commuting with the projections, -linear in each fiber, and preserving the norm. A Banach bundle is sometimes said to be Hilbertizable if it is isomorphic to the underlying Banach bundle of a Hilbert bundle; structurally, there is no difference between a Hilbert bundle and a Hilbertizable Banach bundle (again using the polarisation formula to prove that being a Hilbert space is a property-like structure).
For Banach bundles see ch. 13 in vol. 1 (from page 125; def. 13.4 on p. 127) and for Banach algebraic bundles see from 783 on in vol. 2 of