Exercise 8, § 5

Let F and G be two vector spaces in separating duality. Show that the properties $\alpha)$ and $\beta)$ of IV, p. 52, exerc. 6, a) are also equivalent to the following : \gamma) F, with $\tau(F, G)$ is quasi-complete, and every bounded sequence of points of F has a limit point for $\sigma(F, G)$ (cf. IV, p. 35, th. 1); \delta) F, with $\tau(F, G)$ is quasi-complete, and every decreasing sequence of non-empty closed convex and bounded sets in F has a non-empty intersection (cf. exerc. 6, b)).