Exercise 19, § 4

A Hausdorff locally convex space E is said to be ultrabornological if every convex subset of E which absorbs all the convex, balanced, bounded and semi-complete subsets of E, is a neighbourhood of 0 in E. a) Show that every ultrabornological space is both bornological and barrelled. b) Let E be a Hausdorff locally convex space such that the closed, convex, balanced envelope of the set of points of every sequence tending to 0 is semi-complete. Show that if E is bornological then it is ultrabornological. In particular every bornological and quasi-complete space is ultrabornological; every Fréchet space is ultrabornological.

c) Let $(E_\alpha)$ be a directed increasing family of vector subspaces of a vector space $E$ such that, $E$ is the union of the $E_\alpha$. Let $\mathscr{T}_\alpha$ be a locally convex topology on $E_\alpha$ for every $\alpha$, and let $\mathscr{T}$ be the finest locally convex topology for which the canonical injections from $E_\alpha$ into $E$ are continuous. Suppose that $\mathscr{T}$ is Hausdorff and that, for every $\alpha$, the topology on $E_\alpha$ induced by $\mathscr{T}$ is $\mathscr{T}_\alpha$. Show that if each of the spaces $E_\alpha$ is ultrabornological, then $E$ with $\mathscr{T}$ is ultrabornological.

d) Show that every finite product of ultrabornological spaces is ultrabornological; deduce that every topological direct sum of ultrabornological spaces is ultrabornological.

e) Show that the product space $E = \sum_{n=0}^\infty E_n$ of an infinite sequence of ultrabornological spaces is ultrabornological. (Let $A$ be a convex subset of $E$ which absorbs every convex, balanced, bounded and semi-complete subset of $E$. Show that if $A$ were not a neighbourhood of 0 in $E$, then there would have existed a sequence $(x_n)$ in $A$ such that $x_n$ has its first $n-1$ coordinates zero, but is $\neq 0$. Next observe that the closed convex balanced envelope of the set of points of such a sequence is identical to the set of points $\sum_{n=0}^\infty \lambda_n x_n$, where $\sum_{n=0}^\infty |\lambda_n| \leq 1$, and that this envelope is a semi-complete set.)