Exercise 11, § 5

a) Let E be a Hilbert space and F an everywhere dense subspace of the dual $E'$ of E that is distinct from $E'$; the unit ball B of E is compact for the weak topology $\sigma(E, F)$, and there exists a point $a$ of the unit sphere through which passes no closed (in $\sigma(E, F)$) support hyperplane of B. b) Give E the topology $\sigma(E, F)$ and consider, in the product space $G = E \times \mathbf{R}$ the set A of pairs $(x, \zeta)$ such that $\|x\| < 1, \zeta \geq \|x\|/(1 - \|x\|)$. Show that A is closed and locally compact, but that if D is the line with equation $x = a$ in G then $D \cap A = \varnothing$ and there does not exist any closed hyperplane in G separating A from D. c) Show that, when we give E the topology $\sigma(E, F)$, there exists a continuous affine real valued function in the subspace B of E, that is not the restriction to B of a continuous affine function in E. \*