Exercise 22, § 1

Let E be a Hausdorff locally convex space, $E'$ its dual and M a closed subspace of E. a) Show that if the closed convex envelope of a compact set in E is compact, then the topology of compact convergence on $E'/M^\circ$ (identified with the dual of M) is the quotient topology of the topology of compact convergence on $E'$ by $M^\circ$. b) Show that if M is infrabarrelled and if $E'/M^\circ$, endowed with the topology $\beta(E'/M^\circ, M)$ is bornological, then the topology $\beta(E'/M^\circ, M)$ is the quotient of $\beta(E', E)$ by $M^\circ$.