I am a statistician punching above my weight on a problem involving functional analysis.
I am trying to write down conditions under which I am allowed to apply the chain rule to calculate a derivative of a composition of two functions $F \circ G$. In my problem $G: \mathbb{R}^{\ge 0} \rightarrow L^{\infty}$. The first (only?) part of my difficulties is I'm having a difficult time notating the codomain of $F$. The mapping $F$ can be described as $\psi \mapsto D(\psi)$ for an element $\psi \in L^{\infty}$, where $D(\psi)$ is an element of a Hilbert space $H(\psi)$ with an inner product $\langle \cdot, \cdot \rangle_{\psi}$ that depends on $\psi$.
To add some details about $H(\psi)$: the simplest case is to think of $\psi$ as a probability measure and $H(\psi)$ is the Hilbert space of functions $h$ (appropriately defined on some outcome space) such that $\int h d\psi = 0$ and $\int h^2 d\psi < \infty$, i.e., an $L^2_0(\psi)$ space. Ideally, I'd like to be able to handle slightly more complex cases than this, but this would suffice for now.
I am hoping to argue that the codomain of $F$ is a Banach space, in which case I think I can just assume $F$ and $G$ are Fréchet differentiable and voilá, chain rule. But I am getting hung up on the fact that each Hilbert space above is indexed by the particular $L^{\infty}$ element. So I don't really understand what metric to use in this space or whether indeed the codomain is at all a Banach space.
Again, I'm out of my depths here, so please let me know if I can clarify any elements of the problem.