Good morning,
for research-related reasons I'm very often using the concept of jets of smooth (i.e. $C^\infty$) mappings $\varphi:I\to B$, where $I\subset \mathbb{R}$ is an open interval containing, say, $0$, and $B$ is a Banach or even an Hilbert space (typically $L^1([0,1],\mathbb{R}^d)$ or $L^2([0,1],\mathbb{R}^d)$).
Recently, a colleague of mine asked me where to find a precise treatment of such objects, and since I didn't have any precise reference in mind, I promised him to do some research and come back with an answer. I then started googling but didn't find any precise reference. Even the wikipedia page treats the finite-dimensional case only, and seemingly implies that the situation remains the same in my setting.
This is why I am starting to feel a bit uncomfortable, and want to have a neat reference myself.
Any suggestion will be greatly appreciated. Thank you very much.