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 $\ell^1(\mathbf{N})$ into $\mathbf{R}$, which transforms the bounded sequence of the $e_n$ into an unbounded subset of $\mathbf{R}$ (use Urysohn’s theorem (TG, IX, p. 44, Theorem 2)).