Exercise 9, § 2

a) Let E be an infinite dimensional real hilbertian space satisfying the first axiom of countability. Let $(a_n)_{n \geq 0}$ be a free family of points in E, such that each of the two families $(a_{2n})$ and $(a_{2n+1})$ is total in E (II, p. 80, exerc. 26, $a$). Let F and G be two vector subspaces of E for which $(a_{2n})$ and $(a_{2n+1})$ are respectively (algebraic) bases. The spaces F and G are put in separating duality by the bilinear form $\langle y|z \rangle$. Show that if B denotes the unit ball in E, then in the space F, endowed with the topology $\sigma(F, G)$, the convex set $F \cap B$ is closed, but does not have any closed support hyperplane. b) Let $(b_n)_{n \geq 1}$ be an everywhere dense set in B, and for every $x \in E$, let $u(x)$ be the sequence $(\langle b_k|x\rangle/k)_{k \geq 1}$. Show that $u$ is an injective, continuous linear mapping from E into the hilbertian space $\ell^2(\mathbf{N})$ and that $u(B)$ is compact. Show that, in the normed subspace $L = u(F)$ of $\ell^2(\mathbf{N})$, the set $u(B \cap F)$ is closed, convex and precompact, but does not have any closed support hyperplane (observe that if $f$ is a continuous linear form on L, then $f \circ u$ is a continuous linear form on F for the topology $\sigma(F, G)$).