Exercise 6, § 1

Let F be a distinguished (TVS, IV, §2, Exer. 4 b)) Fréchet space, F' its dual. Show that if $f$ is a scalarly essentially integrable mapping of T into F, then $\int f d\mu$ belongs to the bidual $F''$ of F. (Embed F in $F''$; apply Th. 1, as well as Prop. 2 and the Lemma of the Appendix.)