What is the Frechet derivative of $\cos(x(\cdot))$?
Let $f: C[0,1] \rightarrow \mathbb{R}$, $f(x(\cdot))$.
My approach: by the definition of Frechet derivative we need to consider $$ f((x+h)(t)) - f(x(t)) $$ for $\forall x, h \in C[0,1]$. In our case: $$ \cos((x+h)(t)) - \cos(x(t)) = \sin(x(t))h(t) + o(h) $$
is it correct or I misunderstood something?
Thank you for your help!