Exercise 4, § 5

Let E be a locally convex space, F an infrabarrelled space (III, p. 44, exerc. 7) and $\mathfrak{T}$ the set of all bounded subsets of F. Show that, if a bilinear mapping from $E \times F$ into a locally convex space G is $\mathfrak{T}$-hypocontinuous, it is $(\mathscr{S}, \mathfrak{T})$-hypocontinuous for every set $\mathscr{S}$ of bounded subsets of E ($cf.$ III, p. 44, exerc. 11).