Exercise 11, § 4

a)** Let $E$ be a strict inductive limit of Fréchet spaces $E_n$ satisfying the first axiom of countability, and $F$ an everywhere dense subspace of $E$. For every $n$, put $F_n = \overline{F \cap E_n}$; show that $E$ is the strict inductive limit of the $F_n$ (cf. III, p. 44, exerc. 13, b)). Deduce that $F$ is bornological. (Let $u$ be a linear mapping from $F$ into a Banach space $L$ which transforms every bounded set into a bounded set. Observe that the restriction $u_n$ of $u$ to $F \cap E_n$ is continuous and can be extended to a continuous linear mapping $v_n$ from $\overline{F \cap E_n}$ into $L$, the $v_n$ being restrictions of the same linear mapping $v$ from $E$ into $L$, and conclude that $v$ is continuous.) b) Suppose $F$ is not closed in $E$ but that $F \cap E_n$ is closed in $E_n$ for all $n$ (exerc. 10, b)). Let $A_n$ be a countable dense set in $E_n$, and let $G$ be the vector subspace of $E$ generated by the union of $F$ and the $A_n$. Show that $G$ is a bornological space in which $F$ is a subspace with a complement which has a countable basis, but that $F$ is not infra-barrelled (cf. III, p. 41, § 2, exerc. 4 and p. 45, exerc. 17).