Exercise 19, § 1

Let E and F be two Hausdorff locally convex spaces, E' and F' their duals. For every subset H of the space $\mathscr{L}(E; F)$ of continuous linear mappings from E into F, let 'H denote the set of the transposes of the mapping $u \in H$. For every subset M (resp. N') of E (resp. F'), let H(M) (resp. 'H(N')) denote the union of the sets $u(M)$ (resp. 'u(N')) as u ranges over H. a) For H to be equicontinuous, it is necessary and sufficient that, for every equicontinuous subset N' of F', 'H(N') is an equicontinuous subset of E'. b) Let $\mathfrak{S}$ be a set of bounded subsets of E. Show that H is bounded in $\mathscr{L}(E; F)$ for the $\mathfrak{S}$-topology, if and only if for every $y' \in F'$, 'H(y') is bounded in E' for the $\mathfrak{S}$-topology.

c) Let $\mathfrak{S}$ be a set of bounded subsets of E, $\mathfrak{J}$ a set of bounded subsets of F forming an adapted bornology of F (III, p. 3, def. 4). Suppose $E'$ is assigned the $\mathfrak{S}$-topology and $F'$ the $\mathfrak{J}$-topology. Then $'H$ is equicontinuous if and only if for every set $B \in \mathfrak{S}$, $H(B)$ belongs to $\mathfrak{J}$. In particular, $'H$ is equicontinuous for the strong topologies on $F'$ and $E'$ if and only if $H$ is bounded in $\mathscr{L}(E; F)$ for the topology of bounded convergence. d) Deduce from b) and c) that, if $'H$ is bounded for the topology of simple convergence in $\mathscr{L}(F'; E')$ when $F'$ and $E'$ are assigned the strong topologies, then $'H$ is equicontinuous for these topologies. e) Show that if E is barrelled, the following properties are equivalent : α) $H$ is simply bounded in $\mathscr{L}(E; F)$; β) $H$ is equicontinuous; γ) $'H$ is simply bounded in $\mathscr{L}(F'; E')$ when $E'$ is assigned the weak topology $\sigma(E', E)$; δ) $'H$ is equicontinuous, when $E'$ and $F'$ are assigned the strong topologies. f) Show that if E is infrabarrelled (III, p. 44, exerc. 7), then the properties β) and δ) of e) are equivalent, and are also equivalent to the following : ε) $H$ is bounded in $\mathscr{L}(E; F)$ for the topology of bounded convergence; φ) $'H$ is simply bounded in $\mathscr{L}(F'; E')$ when $E'$ is assigned the strong topology. g) Show that if E is quasi-complete, the properties α), γ) and δ) of e) are equivalent.