Exercise 13, § 5

Let $E$ be a non reflexive Banach space satisfying the first axiom of countability, and let $M$ be a non reflexive closed vector subspace of $E$ of infinite codimension (exerc. 12). Let $(x_n)$ be an everywhere dense sequence in the unit sphere, $C$ the closed convex balanced envelope of the sequence $(x_n/n)$; $C$ is strongly compact in $E$ and $C + M = A$ is a closed convex set (GT, III, § 4, No. 1, cor. 1 to prop. 1). Let $S$ be the unit ball in $E$, and $B = A \cap S$. a) Show that 0 is not an interior point of $A$ and deduce that there exists $x_0 \in E$ such that $\lambda x_0 \notin A$ for $\lambda > 0$. b) Show that there exists no supporting hyperplane of $B$ passing through 0 (observe that such a hyperplane must be a supporting hyperplane of $C$). c) Let $U_0 = M \cap S$, and let $(U_n)_{n \geq 1}$ be a decreasing sequence of closed convex, bounded and non-empty sets, such that $U_1 \subset \frac{1}{2} U_0$ and such that the intersection of the $U_n$ is empty (IV, p. 68, exerc. 8). Let $F$ be the closed convex envelope of the union of the sets $\frac{1}{n} x_0 + U_n$ ($n \geq 1$). Show that $B \cap F = \varnothing$ but that there exists no closed hyperplane separating $B$ and $F$ (use $b$).