Exercise 6, § 2

Let K be the topological division ring obtained by transferring the usual topology of $\mathbf{Q}^2$ to the field $\mathbf{Q}(\sqrt{2})$ by the mapping $(x, y) \mapsto x + y \sqrt{2}$. a) Let E be the set $\mathbf{Q}(\sqrt{2})$ with its vector space structure over K and with the topology induced by that of $\mathbf{R}$. Show that E is a Hausdorff topological vector space, of dimension 1 on K, but that it is not isomorphic to $K_s$. b) Let F be the topological vector space $E \times E$ over K ; in F, the hyperplane $E \times \{0\}$ is closed but there is no continuous linear form $f$ on $E \times E$ such that this hyperplane is given by the equation $f(x) = 0$.