Exercise 10, § 3

Let $E$ be a topological vector space, which is a Baire space, and $F$ be a topological vector space. a) Show that, if a subset $H$ of $\mathscr{L}(E; F)$ is bounded for the topology of simple convergence, then $H$ is equicontinuous (for every closed neighbourhood $V$ of 0 in $F$, consider the sets $M_n = \bigcap_{u \in H} u^{-1}(nV)$). b) Show that, if a subset $H$ of $\mathscr{L}(E, F)$ is not equicontinuous, the set of all $x \in E$ such that $H(x)$ is not bounded in $F$ is the complement of a first category set. Deduce from this that, if $(H_n)$ is a sequence of subsets of $\mathscr{L}(E; F)$ which are not equicontinuous, then there exists an $x \in E$ such that none of the sets $H_n(x)$ is bounded in $F$ (principle of condensation of singularities).