Exercise 10, § 5

Let E be a Hausdorff, quasi-complete and non semi-reflexive locally convex space, and let H be a closed hyperplane in E containing the origin. Let $(C_n)$ be a decreasing sequence of non-empty, closed convex and bounded sets, contained in H and not containing 0 and whose intersection is empty (exerc. 8). Let $x$ be a point not belonging to H, and let A be the closed convex and balanced envelope of the union of the sets $\left(1 - \frac{1}{n}\right)x + C_n$ for $n > 0$. a) Show that there exists no supporting hyperplane of A which is parallel to H (for $y \in x + H$, observe that there exists an integer $n$ such that $y \notin x + C_n$).

b) Let $z \in \mathbf{H}$ be such that $z \notin C_1$; show that the convex envelope of the union of two closed convex and bounded sets $A$ and $B = x + z + C_1$, is not closed (prove that $x + z$ is in the closure of this envelope, but does not belong to it).