Exercise 24, § 2

Let $M$ be an everywhere dense convex set in a topological vector space $E$ (cf. II, p. 66, exerc. 11, c)); show that, for every closed hyperplane $H$ in $E$, the set $H \cap M$ is dense in $H$ (for every point $x_0 \in H$, and every balanced neighbourhood $V$ of 0 in $E$, consider the intersections of $x_0 + V$ and the two open half-spaces determined by $H$, and deduce that $x_0 + V + V$ meets $H \cap M$).