Let $E$ a Banach space, $F=L(E,E)$ of linear and continuous functions. Define $f^0={\rm id}_E$, $f^n=f\circ\cdots\circ f$, $n$ times. Put $\exp(f)=\sum_{n=0}^\infty \frac{f^n}{n!}$. How to show the following two points: Added: $f\in F$ is a linear and continuous function!
- $\exp:F\to F$ is a function of class $C^\infty$.
- $\exp$ is a diffeomorphism of class $C^\infty$ from neighborhood of $0\in F$ to a 'hood of $id_E$.
My approach: I have already proved two facts, $\exp(f)$ is well define by proving the series converges absolutly. And $\|\exp(f)-{\rm id}_E\|<1$ if $\|f\|<\log 2$.
But the other points seems difficult: Because $\exp(f+h)-\exp(f)=\sum_{n=0}^\infty \frac{(f+h)^n}{n!}-\sum_{n=0}^\infty \frac{f^n}{n!}$ and from here nothing is clear to me! I suspect the derivative would be $Dfe^{f}$. But I don't know how to prove it!
Thanks in advance!