Exercise 11, § 2

Let $E_1, E_2$ be two infinite dimensional real hilbertian spaces satisfying the first axiom of countability, and E the hilbertian sum $E_1 \oplus E_2$ (which we shall identify with the product $E_1 \times E_2$). Let $(e_n)_{n \geq 1}$ be an orthonormal basis of $E_1$; in $E_2$, let A be a compact convex set containing 0, and D a line passing through 0, such that $D \cap A = \{0\}$ and suppose there exists no closed supporting hyperplane of A at the point 0 (exerc. 10). Let $(\alpha_n), (\beta_n)$ be two sequences of numbers $\geq 0$ such that $\lim_{n \to \infty} \beta_n = 0$ and $\sum_n \alpha_n^{-1} < 1$. Let P be the set of all points $\sum_n \xi_n e_n$ of $E_1$ such that $0 \leq \xi_n \leq \alpha_n$ for every $n \geq 1$. Finally, let Q be the closed convex envelope in E of the set of all points $(\alpha_n e_n, x + \beta_n a)$, where $n \geq 1, a \neq 0$ is a fixed point in D and x ranges over A. a) Show that $P \cap Q = \varnothing$ and that there exists no closed hyperplane in E separating P and Q. b) Let F be the hilbertian sum $E \oplus \mathbf{R}$, and let c be an arbitrary point of F not contained in E. Show that the pointed convex cones $P_1, Q_1$ with vertex c, generated by P and Q respectively, are closed in F and that there exists no closed hyperplane in F separating $P_1$ and $Q_1$ (to see that $P_1$ and $Q_1$ are closed, prove that neither P nor Q contain the half-line).