Exercise 2, § 5

Let E be the direct sum space $\mathbf{R}^{(\mathbf{N})}$, with the topology induced by the product topology on $\mathbf{R}^{\mathbf{N}}$. Show that the bilinear form $((x_n), (y_n)) \mapsto \sum_{n=0}^{\infty} x_n y_n$ on $E \times E$, is separately continuous, but that for every set $\mathscr{S}$ of bounded subsets of E containing at least one infinite dimensional bounded set, this bilinear form is not $\mathscr{S}$-hypocontinuous.