TITLE: Is this a $C^{\infty}$ function ?
QUESTION [11 upvotes]: Let be $(a_n)\in\ell^2(\mathbb N)$ and consider the mapping $f:\ell^2(\mathbb N)\to\ell^2(\mathbb N)$ given by
$$
f\Big((a_n)\Big)=(a_n^n).
$$
Question: Is $f$ a Fréchet $C^{\infty}$ function in whole $\ell^2(\mathbb N)$ ?
If the answer for the previous question is no. Is there a non-linear $C^{\infty}$ function, defined in some Banach space that maps a closed bounded set onto a non bounded set ?

REPLY [6 votes]: The behaviour of this function is rather more subtle (and interesting) than suggested by the answers so far.  Let me begin with the general remark that the theory of holomorphic functions on Banach spaces (even locally convex spaces) is very well developed---a readable introduction which contains enough material to answer the above questions is a survey article  by Nachbin in vol. 79 of the Bulletin of the American Mathematical Society 
(1973) but a great deal of work has been done since and several monographs have been written on the subject.  The function of the question is indeed holomorphic but only on the open unit ball.   The problem is that in larger balls there are elements with components larger than $1$ and the powers make them huge.  Interestingly, it is holomorphic in weaker senses
everywhere, e.g., it is finitely holomorphic, i.e., its restriction to finite dimensional subspaces is holomorphic (in view of Hartogs' theorem it suffices to check this for one-dimensional spaces).  It is also holomorphic as a function with values in $\omega$, the product of countable copies of the real line. Very mild smoothness properties combined with finite homomorphicity imply holomorphicity in the strong sense and so these cannot hold for the function in question outside of the unit ball.
With regard to the boundedness question, it is a central fact that holomorphic functions on Banach spaces do not necessarily preserve boundedness---in fact, this behaviour is the exception
in a certain sense.  For example, a result quoted in the above-mentioned article states that if a closed, bounded subset of a reflexive or separable space is such that each holomorphic mapping is bounded on it, then it is compact.