Exercise 25, § 1

Let F be a separable Fréchet space in which every Cauchy sequence for $\sigma(F, F')$ is convergent for this topology (for example a space $L^1(T', \nu)$, where $T'$ has a countable base (Ch. V, §5, Exer. 16 b))). Show that every scalarly essentially integrable mapping f of T into F is scalarly well integrable. (First prove that for every function $g \in \mathscr{L}^\infty(\mu)$ with compact support, one has $\int gf\, d\mu \in F$, on observing that f is $\mu$-measurable (No. 5, Prop. 12), and on applying Prop. 8 of No. 2 and the hypothesis. Deduce from this, with the help of the hypothesis, that $\int_A gf\, d\mu \in F$ for every function $g \in \mathscr{L}^\infty$ and every set A that is a countable union of compact sets. Finally, with the notations of Exer. 19, prove that the image under $u_f$ of every equicontinuous subset of $F'$ is relatively weakly compact in $L^1$, using Eberlein’s theorem (TVS, IV, §5, No. 3, Th. 1) and the fact that the equicontinuous subsets of $F'$ are metrizable for $\sigma(F', F)$.)