Banach bundle

Definition

Let $M$ be a Banach manifold of class $C^p$, with $p \ge 0$. Let $E$ be a topological space and $\pi \colon E \to M$ a continuous surjection, and assume each fibre $E_x = \pi^{-1}(x)$ carries the structure of a real Banach space. For Banach spaces $X$ and $Y$, let $\mathcal{L}(X; Y)$ be the Banach space of bounded linear maps with the operator norm. The map $\pi$ is a Banach bundle of class $C^p$ if there are data of the following kind. There are an open cover $(U_i)_{i \in I}$ of $M$, Banach spaces $X_i$ and maps $\tau_i \colon \pi^{-1}(U_i) \to U_i \times X_i$ such that:

1. Each $\tau_i$ is a homeomorphism with $\mathrm{pr}_{U_i} \circ \tau_i = \pi$.
2. For each $x \in U_i$, the induced map $\tau_{i,x} \colon E_x \to X_i$ is a continuous linear isomorphism.
3. For all $i, j \in I$, the map $U_i \cap U_j \to \mathcal{L}(X_i; X_j)$, $x \mapsto \tau_{j,x} \circ \tau_{i,x}^{-1}$, is of class $C^p$.