Let $X$ be an infinite-dimensional Banach space, and let $B=\{x\in X: \|x\|\leq 1\}$ be its closed unit ball.
Does there exist a continuous mapping $F: X\to X$ such that the set $F(B)=\{F(x): x\in B\}$ is unbounded?
Of course, such $F$ cannot be a linear operator, so $F$ must be nonlinear.
Thank you very much in advance!
