Lemma 3

Let F be a Hausdorff locally convex space, $f$ an element of $\mathscr{L}_{F'_s}^\infty$. For every $z \in F$, one has $\langle z, f \rangle \in \mathscr{L}^\infty$, and the linear mapping $z \mapsto \pi(\langle z, f \rangle)$ of F into $L^\infty$ is continuous; if, moreover, F is a normed space, then $N_\infty(\langle z, f \rangle) \leq |z| \cdot \sup_{t \in T} |f(t)|$.

It is clear by definition that $\langle z, f \rangle$ is $\mu$-measurable and bounded in measure; replacing if necessary $f$ by a function belonging to the same class of $L_{F'_s}^\infty$, we can suppose in addition that $f(T) \subset V^\circ$, where V is a balanced convex neighborhood of 0 in F (which does not modify $\langle z, f \rangle$ except on a locally negligible set, depending on $z$). Then the relation $z \in V$ implies that $|\langle z, f(t) \rangle| \leq 1$ for all $t \in T$, which proves the continuity of $z \mapsto \pi(\langle z, f \rangle)$. The final assertion is obvious.