Exercise 8, § 2

Let $p$ be a prime number, $\mathbf{Q}_p$ the field of $p$-adic numbers (GT, III, § 6, exerc. 23). Let $E_0$ be the topological product space $\mathbf{Q}_p \times \mathbf{R}$; if K denotes the field $\mathbf{Q}$ with the discrete topology, then $E_0$ is a topological vector space over K. Let M be the vector subspace formed by the elements $(r, r)$ where $r$ varies in $\mathbf{Q}$; further let $\theta$ be an irrational number and N the vector subspace formed by the elements $(0, r\theta)$ where $r$ varies in $\mathbf{Q}$. Let E be the subspace $M + N$ of $E_0$; show that N is a closed hyperplane in E, but that there does not exist a complementary topological subspace to N (note that M is everywhere dense in $E_0$).