Exercise 15, § 1

a)** Let E be a locally convex Hausdorff space, A a convex and balanced set in E, and u a linear form on E. Show that if $A \cap u^{-1}(0)$ is closed with respect to A, then the restriction of u to A is continuous. (If not, show that 0 will be in the closure of the intersection of A and a hyperplane $u^{-1}(\alpha)$ with $\alpha \neq 0$; deduce that if $b \in A$ is such that $u(b) = -\alpha$, the point $\frac{1}{2} b$ will be in the closure of $A \cap u^{-1}(0)$.) b) Let E be a real locally convex Hausdorff and complete space, E' its dual. Show that if a hyperplane H' of E' is such that its intersection with every equicontinuous and weakly closed subset $M' \subset E'$ is weakly closed, then H' is weakly closed (use a) and III, p. 21, cor. 1). c) Let E be a locally convex Hausdorff space, E' its dual and C a convex, balanced and closed set in E. Let u be a linear form on E; show that if the restriction of u to C is continuous for the initial topology, it is also continuous for $\sigma(E, E')$ (use a). Show by an example that the restriction of u to the vector subspace M generated by C is not necessarily continuous (take $E = \mathbf{R}^{(N)}$ with the norm $\|x\| = \sup_n |\xi_n|$, and for C take a suitable convex set generating E).