Exercise 5, § 5

a) Let E, F and G be three Hausdorff locally convex spaces, and $u$ a bilinear mapping from $E \times F$ into G. In order that there exist a balanced neighbourhood U of 0 in E such that the set of all mappings $u(x, .)$, where x runs through U, is equicontinuous in $\mathscr{L}(F; G)$, it is necessary and sufficient that $u$ is continuous when we replace the topology of E by the coarsest topology for which the sets $\lambda U$ ($\lambda \neq 0$) form a fundamental system of neighbourhoods of 0. Show that if G is normed, this condition is satisfied by every continuous bilinear mapping from $E \times F$ into G. b) Take for E, F and G the product space $\mathbf{R}^{\mathbf{N}}$, and for $u$ the continuous bilinear mapping $((x_n), (y_n)) \mapsto (x_n y_n)$. Show that there does not exist any neighbourhood U of 0 in E such that the set of maps $u(x, .)$, where x runs through U, is equicontinuous in $\mathscr{L}(F; G)$.