Exercise 4, § 5

Let E be the Banach space $\ell$^∞(\mathbf{N}). which does not satisfy the first axiom of countability (I, p. 25, exerc. 1), and let E' be its dual. For every integer n ≥ 0, let e'_n be the continuous linear form on E which associates to each $x = (\xi_n) \in E$ the nth term of this sequence. Show that the sequence $(e'_n)$ is total in $E'$ for $\sigma(E', E)$; moreover, every sequence extracted from $(e'_n)$ has a limit point in $E'$, for the topology $\sigma(E', E)$, but there exists no sequence extracted from $(e'_n)$ which converges in $E'$ for this topology.