Exercise 17, § 1

Let E be a Hausdorff locally convex space, E' its dual and, in E', let $\mathfrak{B}_1$ be the family of convex equicontinuous sets, $\mathfrak{B}_2$ the family of convex, relatively weakly compact subsets, $\mathfrak{B}_3$ the family of convex strongly bounded subsets, and $\mathfrak{B}_4$ the family of convex weakly bounded subsets. Then $\mathfrak{B}_1 \subset \mathfrak{B}_2 \subset \mathfrak{B}_3 \subset \mathfrak{B}_4$. Give an example of a space E for which the four families of sets are distinct (take for E a product of three spaces for which we have, respectively $\mathfrak{B}_1 \neq \mathfrak{B}_2$ (cf. IV, p. 49, exerc. 8), $\mathfrak{B}_2 \neq \mathfrak{B}_3$ (exerc. 16, d)), $\mathfrak{B}_3 \neq \mathfrak{B}_4$ (III, p. 23, Remark 2).