Exercise 3, § 5

Let E be the space $\mathbf{R}^{(\mathbf{N})}$ with the finest locally convex topology (II, p. 26); let F be the space $\mathbf{R}^{\mathbf{N}}$; the space E is ultrabornological (III, p. 45, exerc. 19) and complete, whilst F is metrizable and complete. Let $\mathscr{S}$ (resp. $\mathfrak{T}$) be the set of all bounded subsets of E (resp. F). Show that the bilinear form $((x_n), (y_n)) \mapsto \sum_{n=0}^{\infty} x_n y_n$ on $E \times F$ is $(\mathscr{S}, \mathfrak{T})$-hypocontinuous, but is not continuous ($cf.$ IV, p. 48, exerc. 11).