Exercise 3, § 1

Let F be a vector space. Show that, for the topology σ(F, F*), every bounded subset of F is finite dimensional. Deduce that, if F is infinite dimensional, there exist, in the completion $\hat{F}$ of F (for σ(F, F*)), compact subsets which are not contained in the closure of any bounded subset of F (cf. II, p. 52, prop. 10).