Exercise 22, § 1

In a Hausdorff locally convex space $F$, let $(z_\alpha)_{\alpha \in A}$ be a family such that the mapping $\alpha \mapsto z_\alpha$ is scalarly well integrable (Exer. 19) when $A$ is equipped with the measure defined by mass +1 at each point. a) Show that the family $(z_\alpha)$ is summable for every topology $\mathscr{T}$ compatible with the duality between $F$ and $F'$ (make use of Exer. 19 a)). Converse when $F$ is quasi-complete for $\mathscr{T}$. b) Suppose that $F = G'$, $F' = G$, where $G$ is a Hausdorff locally convex space such that there exists in $G'$ a countable strongly dense subset. Show that $(z_\alpha)$ is then summable for the strong topology on $G'$ (which is not necessarily compatible with the duality between $G'$ and $G$ (cf. Exer. 19d)). Show that this result does not extend to the case that $G = L^1(\mathbf{N})$, $G' = L^\infty(\mathbf{N})$.