TITLE: Continuity in Banach space for non-linear maps
QUESTION [5 upvotes]: I want to find an example of a Banach space $X$ and a continuous map $f:X\rightarrow X$ such that $f$ is not bounded on the unit ball. I do not doubt that such an example exists, but I cannot make it explicit.

REPLY [5 votes]: For $X=\ell^1$, put $f(x)=\sum_{n\ge 1} nx_i^n$ (this one is even analytic if you understand that word in a not too restrictive sense).