Exercise 21, § 1

a) Let E be a vector space, $E^*$ its algebraic dual. Show that the topology $\tau(E, E^*)$ is the finest locally convex topology and that the topology $\tau(E^*, E)$ is identical with $\sigma(E^*, E)$. b) Let $E^{**}$ be the algebraic dual of $E^*$. Show that if E is infinite dimensional, then E is dense in $E^{**}$ for all the topologies compatible with the duality between $E^{**}$ and $E^*$, but that the topology induced on E by $\tau(E^{**}, E^*)$ is distinct from $\tau(E, E^*)$.