Exercise 8, § 1

Let $E$ be an infinite dimensional locally convex metrizable space, and $E'$ its dual. In order that the topology $\tau(E, E')$ be identical with the weakened topology $\sigma(E, E')$ it is necessary and sufficient that $E$ is isomorphic to an everywhere dense subspace of the product space $\mathbf{R}^N$ (resp. $\mathbf{C}^N$). (Observe that in $E'$, for the bornology consisting of equicontinuous sets, there exists a countable base (III, p. 1) consisting of finite dimensional sets; conclude that the vector space $E'$ has a countable basis).

Give an example of a locally convex Hausdorff space $E$ for which the initial topology, and the topologies $\sigma(E, E')$ and $\tau(E, E')$ are all distinct.