Continuity in Banach space for non-linear maps 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.
 A: Find a sequence of disjoint balls $(B_n)$, all contained in the unit ball of $X$ and all having radius greater than some fixed $\epsilon$. Select continuous functions $f_n: B_n \to X$ however you like, such that each $f_n$ is zero outside radius $\epsilon/2$. Then patch together to get a continuous function on $\bigcup B_n$, and extend by zero elsewhere to get a function from $X$ to $X$. There needn't be a uniform bound on the $f_n$ for this to work.
A: For a somewhat less explicit example, it suffices to find a continuous $g : X \to \mathbb{R}$ which is unbounded on the unit ball (then fix any nonzero $x_0 \in X$ and take $f(x) = g(x) x_0$).  Pick a discrete subset $\{x_n\}$ from the unit ball (use the Riesz lemma) and set $g(x_n) = n$.  Invoke the Tietze extension theorem to extend $g$ to all of $X$.
A: 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).
A: On $c_0$ or $\ell^\infty$, let
$$f(x)_j =  j \max(0, x_j - 1/2 - \sup_{i\ne j} x_i)$$
