Exercise 21, § 1

Let $f$ be a scalarly well integrable mapping of $T$ into a quasi-complete Hausdorff locally convex space $F$. a) Show that the image under $g \mapsto \int gf\, d\mu$ of the unit ball of $L^\infty$ is a relatively compact subset of $F$ for the original topology, in each of the following cases: $1^\circ$ there exists a countable dense subset of $F$; $2^\circ$ $F$ is the dual $G'$ of a space $G$ that is either metrizable or is the strict direct limit of a sequence of metrizable spaces, and $F$ is equipped with the topology $\tau(G', G)$. (Make use of Exer. 19 c) and Exer. 20; in case $2^\circ$, use Smulian’s theorem (TVS, IV, §5, Exer. 2 c)).) b) Show that the conclusion of a) remains valid when, with no new hypothesis on $F$, one assumes $f$ to be measurable. (Embedding $F$ in a product of Banach spaces, reduce to the case that $F$ is a Banach space; then, using Exer. 19 a), reduce to the case that $T$ is compact; then apply case $1^\circ$ of a).)