Exercise 20, § 2

Let E be a Banach space, E' its dual, and B the unit ball in E. In order that every point of B have a countable fundamental system of neighbourhoods for the topology induced by the weakened topology $\sigma(E, E')$ on B, it is necessary and sufficient that E' satisfies the first axiom of countability for the strong topology. (To show that the condition is necessary, observe that if every point in B has a countable fundamental system of neighbourhoods for the weakened topology, then this is also true for the closure $B^{\circ\circ}$ of B in E'' for the topology $\sigma(E'', E')$. Therefore there exists a sequence $(a'_n)$ in E' such that every neighbourhood of 0 in $B^{\circ\circ}$ for $\sigma(E'', E')$ contains the intersection of $B^{\circ\circ}$ and of a finite number of polars $\{a'_n\}^\circ$; consider the strongly closed subspace W' of E' generated by the $a'_n$, and the orthogonal ${W'}^\circ$ of W' in E''.)