Puzzling problem about differential calculus on an arbitrary normed vector space

Question

Given two normed vector spaces $E$ and $F$, let $U$ be an open set of $E$, and define the application $T:U\rightarrow L_c(E,F)$ of class $C^1$, where $L_c(E,F)$ is the space of linear and continuous applications from $E$ to $F$. We set $f:U\rightarrow F$, the application defined of $U$ such that $f(x)=T(x)(x)$ for all $x$ in $U$.

1. Show that $f$ is differentiable (using the definition), and compute $Df(a)$ for all a in U.
2. Deduce that $f$ is of class $C^1$.