Exercise 11, § 4

Let I be an infinite non-enumerable set. On the vector space $E = \mathbf{R}^{(I)}$, show that the finest locally convex topology is distinct from the topology $\mathscr{T}_0$ defined in I, p. 24, exerc. 14; for this prove that the set of the $x = (\xi_i) \in E$ such that $\left| \sum_{i \in I} \xi_i \right| < 1$, is open in $\mathscr{T}$ but not in $\mathscr{T}_0$.