Exercise 22, § 4

Let $E$ be an infinite dimensional Banach space satisfying the first axiom of countability. a) Show that the set $\mathscr{K}$ of all compact, convex and balanced subsets $A$ of $E$ such that $E_A$ is infinite dimensional, is infinite and has a cardinality $\leq 2^{\mathrm{Card}(\mathbf{N})}$ (GT, IX, § 5, exerc. 17). For every $x_0 \in E$ and every $A \in \mathscr{K}$, the set $x_0 + A$ contains a free subset of cardinality $2^{\mathrm{Card}(\mathbf{N})}$ (II, p. 80, exerc. 24, c)). b) Let $x_0 \neq 0$ be in $E$. Show that there exists a family $(y_A)_{A \in \mathscr{K}}$ such that $x_0$ and the $y_A$ form a free family and that we have $y_A \in x_0 + A$ for all $A \in \mathscr{K}$ (well order $\mathscr{K}$ and argue by transfinite induction, using $a$). c) Let $f \in E^*$ be a linear form such that $f(x_0) = 1$ and $f(y_A) = 0$ for all $A \in \mathscr{K}$, and let $H = f^{-1}(0)$. Show that a subset $M$ of $H$ which is convex, balanced and semi-complete is necessarily finite dimensional (observe that if not, $M$ will contain an infinite dimensional compact convex and balanced set $A$; hence $y_A$ will belong to $H \cap (x_0 + M)$). d) Show that $H$ with the topology induced by that of $E$ is not ultrabornological, in spite of being bornological and barrelled (III, p. 44, exerc. 14). (By using $c$, show that if $H$ were ultrabornological, its topology would have been the finest locally convex topology, and deduce a contradiction.)