Exercise 17, § 6

Let E be an infinite dimensional vector space. a) Show that there exist hyperplanes in $E^*$ that are everywhere dense for the topology $\sigma(E^*, E)$. b) If $H'$ is such a hyperplane, show that, in E, the only linear subvarieties $\neq E$ that are everywhere dense for the topology $\sigma(E, H')$ are the hyperplanes.