Corollary 2

*Let $G$ be a Hausdorff locally convex space having the property (GDF), and $H$ a semi-reflexive space whose strong dual $H_b'$ has the property (GDF) (cf. App., No. 2, Prop. 3). Let $F$ be the space $\mathscr{L}_s(G; H)$; for every mapping $U$ of $T$ into $F$ that is scalarly essentially $\mu$-integrable, the integral $\int U d\mu$ belongs to $F$.

Since $G$ is barreled (Prop. 11), $\mathscr{L}(G; H) = \mathscr{L}(G_\sigma; H_\sigma)$ (TVS, IV, §1, No. 3, Prop. 7); moreover, one can replace $F = \mathscr{L}_s(G; H)$ by the space $\mathscr{L}_s(G_\sigma; H_\sigma)$, since the two spaces have the same dual $G \otimes H'$ (TVS, II, §6, No. 2, Prop. 3, and the first paragraph of No. 3 above). If, for every $u \in \mathscr{L}(G; H) = \mathscr{L}(G_\sigma; H_\sigma)$, one sets $\tilde{u}(x, y') = \langle u(x, y') \rangle$ (for $x \in G$, y' \in H' ), the linear mapping $u \mapsto \tilde{u}$ is a bijection of F onto the space $F_1$ of separately continuous bilinear forms on $G_\sigma \times H'_s$, where $H'_s$ denotes the dual $H'$ equipped with the weak topology $\sigma(H', H)$ (App., No. 1); moreover, this mapping is an isomorphism of $\mathscr{L}_s(G_\sigma; H_\sigma)$ onto $F_1$ equipped with the topology of pointwise convergence (loc. cit.). But since by hypothesis H is the dual of $H'_b$, $F_1$ is also the space of separately continuous bilinear forms on $G \times H'_b$. Cor. 2 therefore follows from Cor. 1.

Note that Cor. 2 is applicable in particular when G is Banach space and H is a reflexive Banach space.