bundle’s base or fibres. Below we point out some peculiarities of the case of a Hilbert bundle whose fibres are generally infinitely dimensional.

\[Defn3.2\] A Hilbert fibre bundle is a fibre bundle whose fibres are homeomorphic Hilbert spaces or, equivalently, whose (standard) fibre is a Hilbert space.

In the present investigation we shall show that the Hilbert bundles can be taken as a natural mathematical framework for a geometrical formulation of quantum mechanics. For linear transports in a Hilbert bundle are valid all results of [@bp-normalF-LTP; @bp-LTP-general; @bp-TP-general] with a possible exception of the ones in which (local) bases in the fibres are involved. The cause for this is that the dimension of a Hilbert space is (generally) infinity. So, there arise problems connected with the convergence or divergence of the corresponding sums or integrals. Below we try to avoid these problems and to formulate our assertions and results in an invariant way.

Of course, propositions \[Prop3.1\] and \[Prop3.2\] remain valid on Hilbert bundles; the only addition is that the vector spaces $V$ and ${{}\mspace{3mu}{\vphantom{V}}    _{}^{\star}\mspace{-0.8mu}V}$ are now Hilbert spaces.

Below, in Sect. \[new-I\] (see below the paragraph after equation [[(\[4.12f\])]{}]{}), we shall establish a result specific for the Hilbert 