Corollary 1

*Let $F$ be a Hausdorff locally convex space containing a countable dense subset. For every function $f \in \mathscr{L}_{F'}^\infty$, every $z \in F$ and every function $g \in \mathscr{L}^1$, let $\Phi_f(z, \tilde{g}) = \int \langle z, f(t) \rangle g(t)\, d\mu(t)$. The mapping $f \mapsto \Phi_f$ defines, by passage to the quotient, a linear bijection of $L_{F'}^\infty$ onto the space $\mathscr{B}(F, L^1)$ of continuous bilinear forms on $F \times L^1$. If $F$ is a normed space, this bijection is an isometry.

One can suppose that $f(T)$ is an equicontinuous subset of $F'$. It is then clear that $\Phi_f$ is separately continuous, and, with the notations of Th. 1 and the Appendix, one has (taking into account the fact that $L^\infty$ is the dual of $L^1$ (Ch. V, §5, No. 8, Th. 4)) $^l \Phi_f = v_f$. The corollary then follows from Th. 1 above and from the Appendix, No. 1, Prop. 1 and its corollary.