Exercise 10, § 2

Let E be an infinite dimensional real hilbertian space satisfying the first axiom of countability, and $(e_n)_{n \geq 1}$ an orthonormal basis of E. a) Let A be the closed convex balanced envelope of the set of all points $e_n/n$ in E. Show that A is compact and that there does not exist any closed supporting hyperplane of A at the point 0, but that there exist lines D passing through 0 such that $D \cap A = \{0\}$. b) Let F be the hilbertian sum $E \oplus \mathbf{R}$, $e_0$ a vector which, with the $e_n$ (for $n \geq 1$) forms an orthonormal basis of F. If B is the closed convex envelope of $\{e_0\} \cup A$, show that there exists a closed segment L with mid-point 0 in F, such that $L \cap B = \{0\}$, but there does not exist any closed hyperplane passing through 0 which separates L and B (even though there exists a closed supporting hyperplane of B at 0).