Corollary

— Let $\mu$ be a positive measure on $X$, $F$ a Banach space, $F'$ its strong dual, and $g$ any function in $\mathscr{L}^q_{F'}$. Then, the linear form on $L^p_F$, deduced from the linear form $f \mapsto \int \langle f, g \rangle d\mu$ on $\mathscr{L}^p_F$ by passage to the quotient, is continuous and has norm $N_q(g)$.