Lemma 2

Let f be a mapping of T into $F'_s$, such that for every $z \in F$, the numerical function $\langle z, f \rangle$ belongs to $\overline{\mathscr{L}}^p(\mu)$ ($1 \leq p \leq +\infty$); let $\theta(z)$ be the class of this function in $L^p(\mu)$. Then $z \mapsto \theta(z)$ is a continuous linear mapping of F into $L^p(\mu)$.

By virtue of the (GDF) property, it suffices to show that for every sequence $(z_n)$ of elements of F converging to $z$, such that $(\theta(z_n))$ converges to $u \in L^p(\mu)$, one has $u = \theta(z)$. Now, replacing if necessary the sequence $(z_n)$ by a subsequence, one can suppose that the sequence of functions $\langle z_n, f \rangle$ converges locally almost everywhere to a function $h \in \overline{\mathscr{L}}^p(\mu)$, with class $u$ in $L^p(\mu)$ (Ch. IV, §3, No. 4, Th. 3 and Ch. V, §1, No. 3). Since, by hypothesis, for every $t \in T$ the sequence $(\langle z_n, f(t) \rangle)$ converges to $\langle z, f(t) \rangle$, we have $h(t) = \langle z, f(t) \rangle$ locally almost everywhere, consequently $u = \theta(z)$.