Banach-Bundle: What is differentiability on "only" topological space?

Question

I have a question concerning the definition given in this Wikipedia article. There, a banach-bundle is defined. My question refers to the second bullet point. There, the map $\tau_{ij}: U_i \cap U_j \to \text{Lin}(X_i,X_j)$ has to be a differentiable map of $C^p$. But the $U_i$ and $U_j$ are just open subsets of the banach-manifold $X$. As far as I know, we only get that $X$ is a topological space, not however a normed vector space. So, how can the map $\tau_{ij}$ be a differential map if the domain is not a normed vector space? Is there some way to identify $U_i \cap U_j$ with a normed vector space? Or do we talk about some other differentiability?