transports along paths and\
Hilbert fibre bundles {#III}
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The general theory of linear transports along paths in vector bundles is developed at length in [@bp-normalF-LTP; @bp-LTP-general]. In the present investigation we shall need only a few definitions and results from these papers when the bundle considered is a Hilbert one (see below definition \[Defn3.2\]). To their partial introduction and motivation is devoted the current section.

Let $(E,\pi,B)$ be a complex[^8] vector bundle  with bundle (total) space $E$, base $B$, projection $\pi\colon E\to B$, and isomorphic fibres $\pi^{-1}(x)\subset E$, $x\in B$. Let $\mathcal{E}$ be the (standard, typical) fibre of the bundle, [i.e. ]{}a vector space to which all $\pi^{-1}(x)$, $x\in B$ are homeomorphic (isomorphic). By $J$ and $\gamma\colon J\to B$ we denote, respectively, a real interval and path in $B$.

\[Defn3.1\] A linear transport along paths in the bundle $(E,\pi,B)$ is a map $L$ assigning to any path $\gamma$ a map $L^\gamma$, transport along $\gamma$, such that $L^\gamma\colon (s,t)\mapsto L^\gamma_{s\to t}$ where the map $$\label{3.fibre}
L^\gamma_{s\to t} \colon  \pi^{-1}(\gamma(s)) \to \pi^{-1}(\gamma(t))
    \qquad s,t\in J,$$ called transport along $\gamma$ from $s$ to $t$, has the properties: $$\begin{aligned}
{2}                  \label{3.1}
L^\gamma_{s\to t}\circ L^\gamma_{r\to s} &=
            L^\gamma_{r\to t},&\qquad  r,s,t&\in J, \\
L^\gamma_{s\to s} 