Remark 5

— In general the bounded subsets of a locally convex space $E$ are not all precompact (for example, if $E$ is an infinite dimensional normed space, its unit ball is not compact (I, p. 15, th. 3)). However, this is so if $E$ is a weak space (II, p. 42) : for the Hausdorff topological vector space associated with E is then isomorphic to a subspace of a product K^l whose bounded subsets are precompact (cf. III, p. 4, cor. 2). For other examples, see IV, p. 18.