Corollary 3

— In a locally convex space, the closure of each linear variety $M$ is the intersection of the closed hyperplanes that contain $M$.

For all $x \notin \overline{M}$, let $H$ be a closed hyperplane that separates $x$ strictly from $\overline{M}$;

thus $\overline{M}$ is parallel to $H$; the closed hyperplane $H_1$, containing $\overline{M}$ and parallel to $H$ does not contain $x$. The corollary follows.