Let consider $\xi:\mathcal{L}_c(E)\to\mathcal{L}_c(E), \ f \mapsto \sum_{n\ge 0}\frac{f^n}{n!}$ where $E$ is a Banach space.
I have to prove that $\xi$ is differentiable on $\mathcal{L}_c(E)$.
The suggestion is to use the following theorem :
Let $O$ a bounded convex set of $E$ (a normed vector space) and $F$ a Banach space. Let $(g_n)_{n\ge 0}$ a sequence of differentiable functions from $O$ to $F$.
If $(\mathrm{d}g_n)_{n\ge 0}$ converges uniformly on $O$ to $h : O \to \mathcal{L}(E,F)$ and if there exists $x_0 \in O$ such that $(g_n(x_0))$ converges.
Then $(g_n)_{n\ge 0}$ converges to $g : O\to F$ with $g$ a differentiable function on $O$ and $\mathrm{d}g=h$.
I do not know how to start. Do I have to consider the partial sum of the series and adapt the theorem ?
Maybe I have to consider the sequence $\xi_n =\frac{f^n}{n!}$ ?
Thanks in advance !