Exercise 9, § 4

Let I be an uncountably infinite set; on the direct sum vector space $E = \mathbf{R}^{(I)}$, consider, on the one hand, the finest locally convex topology $\mathscr{T}$, and on the other hand, the topology $\mathscr{T}_0$ defined in I, p. 24, exerc. 14, which is locally convex; we know that $\mathscr{T}$ and $\mathscr{T}_0$ are distinct (II, p. 75, exerc. 11) and that E with $\mathscr{T}_0$ is complete (GT, III, § 3, exerc. 10). Show that the bounded sets in E are the same for $\mathscr{T}$ and $\mathscr{T}_0$ (III, p. 39, exerc. 10) and that E with $\mathscr{T}_0$ is not barrelled (observe that the set T of all $x = (\xi_i) \in E$ such that $\sum_{i \in I} |\xi_i| \leq 1$ is a barrel and use exerc. 11 of II, p. 75).