Exercise 20, § 4

Show that, for a Hausdorff locally convex space $E$ to be ultrabornological it is necessary and sufficient that it is the inductive limit of a family of Banach spaces. (To see that the condition is necessary, consider the convex, balanced, bounded and semi-complete sets $B$ in $E$, and the spaces $E_B$. To see that it is sufficient, observe that if $E$ is the inductive limit of a family of Banach spaces $E_\alpha$, we can assume that the $E_\alpha$ are (algebraically) subspaces of $E$; if $V$ is a convex set in $E$ which absorbs the convex, balanced, bounded and semi-complete subsets of $E$, show that $V$ absorbs each ball $B_\alpha$ of $E_\alpha$ (argue by reductio ad absurdum); if $V$ does not absorb $B_\alpha$, then it does not absorb a sequence $(x_n)$ of points of $B_\alpha$, tending to 0 in $E$; then use the fact that in a Banach space, the closed, convex envelope of a compact set is compact.)