This might be a trivial question but I am not very familiar with the subject matter. I was wondering if some sort of mean value theorem works for operators on function spaces. Say $F: \mathcal{S_1} \to \mathcal{S_2}$ is an operator on the function spaces $\mathcal{S_{1,2}}$ then for every $f,g \in \mathcal{S_1}$ there exist $h$ such that \begin{align*} F(f) - F (g) = [D(h)] DF(h)] (f - g), \end{align*} or something like that! What is a good reference to look at if I want to learn more.
This might be a trivial question but I am not very familiar with the subject matter. I was wondering if some sort of mean value theorem works for operators on function spaces. Say $F: \mathcal{S_1} \to \mathcal{S_2}$ is an operator on the function spaces $\mathcal{S_{1,2}}$ then for every $f,g \in \mathcal{S_1}$ there exist $h$ such that \begin{align*} F(f) - F (g) = [D(h)] (f - g), \end{align*} or something like that! What is a good reference to look at if I want to learn more.