Theorem 1

(Gelfand–Dunford)—*Let F be a Hausdorff locally convex space having the property (GDF), $F'_s$ its weak dual. For every mapping f of T into $F'_s$ that is scalarly essentially $\mu$-integrable, the integral $\int f d\mu$ belongs to $F'$.*

Recall that the dual of $F'_s$ is F (TVS, II, §6, No. 2, Prop. 3). For every $z \in F$, the numerical function $\langle z, f \rangle$ is therefore essentially $\mu$-integrable; let $\theta(z)$ be its class in $L^1(\mu)$. To show that $\int f d\mu \in F'$, one must establish that the linear form $z \mapsto \langle z, \int f d\mu \rangle$ is continuous on F; in fact, we are going to prove the following stronger result: