Exercise 18, § 1

In the Banach space $\ell^1(\mathbf{N})$ (I, p. 4) let $e_m$ be the sequence $(\delta_{mn})_{n \geq 0}$ such that $\delta_{mn} = 0$ for $m \neq n$ and $\delta_{nn} = 1$. Define a continuous mapping from $\ell^1(\mathbf{N})$ in $\mathbf{R}$ which transforms the bounded sequence of the $e_n$ into a non bounded subset of $\mathbf{R}$ (use Urysohn's th. (GT, IX, § 4, No. 2, th. 2)).