Theorem 1

— Let F be a Hausdorff locally convex space that contains a countable dense subset. For every function $f \in \mathscr{L}_{F'_s}^\infty$ and every $z \in F$, let $v_f(z) = \pi(\langle z, f \rangle) \in L^\infty$; the mapping $f \mapsto v_f$ defines, by passage to the quotient, a linear bijection of $L_{F'}^\infty$ onto the space $\mathscr{L}(F; L^\infty)$ of continuous linear mappings of $F$ into $L^\infty$. If $F$ is a normed space, this bijection is an isometry.

In view of Lemma 3, the first assertion will be demonstrated if one proves that for every continuous mapping $u$ of $F$ into $L^\infty$, there exists a function $f \in \mathscr{L}_{F'}^\infty$ such that $\pi(\langle z, f \rangle) = u(z)$ for all $z \in F$, and that the class of $f$ in $L_{F'}^\infty$ is uniquely determined by this condition. The second point is immediate, because if $\pi(\langle z, f \rangle) = \pi(\langle z, f_1 \rangle)$ for all $z \in F$, then $f_1 - f$ is scalarly locally negligible. On the other hand, there exists a lifting $\rho$ of $u(F)$ into $\mathscr{L}^\infty$ (Lemma 2). For every $t \in T$, the mapping $z \mapsto \rho(u(z))(t)$ is a continuous linear form on $F$, that is, an element $f(t)$ of $F'$. The function $f$ is scalarly $\mu$-measurable since $\langle z, f \rangle = \rho(u(z)) \in \mathscr{L}^\infty$ for every $z \in F$; one has $\pi(\langle z, f \rangle) = u(z)$; finally, for every $t \in T$ and every $z$ belonging to the inverse image $V$ under $u$ of the unit ball of $L^\infty$,

which shows that $f(t) \in V^\circ$ for all $t \in T$.

If, moreover, $F$ is a normed space, the foregoing shows that

But on the other hand (Lemma 3), $N_\infty(u(z)) \leq |z| \cdot \sup_{t \in T} |f(t)|$, and this inequality continues to hold when $f$ is modified arbitrarily on a locally negligible set. It follows that $\|u\| = N_\infty(|f|)$.