Proposition 12

*Let m be a complex measure on T, p and q conjugate exponents (Ch. IV, §6, No. 4). The bilinear form $(f,g) \mapsto \int fg\, dm$ is defined and continuous on the product $\mathscr{L}_C^p(m) \times \mathscr{L}_C^q(m)$; the inequality $|\int fg\, dm| \leq N_p(f)N_q(g)$ holds, and $N_q(g)$ is the norm of the continuous linear form on $\mathscr{L}_C^p(m)$ deduced from the linear form $f \mapsto \int fg\, dm$ by passage to the quotient.

Moreover, if $1 \leq p < +\infty$, then every continuous linear form on the complex vector space $\mathscr{L}_C^p(m)$ is of the type $f \mapsto \int fg\, dm$, where $g$ is a function in $\mathscr{L}_C^q(m)$, whose class in $\mathscr{L}_C^q(m)$ is uniquely determined.

Since $m = h \cdot |m|$, where $|h(t)| = 1$ locally almost everywhere, the first assertion follows at once from Hölder’s inequality (Ch. IV, §6, No. 4, Th. 2); the second derives from Prop. 3 of Ch. IV, §6, No. 4. Finally, if $u$ is a continuous linear form on $\mathscr{L}_C^p$, its restriction to the (real) subspace $\mathscr{L}^p$ of $\mathscr{L}_C^p$ is a continuous $\mathbf{R}$-linear mapping of $\mathscr{L}^p$ into $\mathbf{C}$; if $1 \leq p < +\infty$, it is therefore of the type $f \mapsto \int fg_1\, d|m| + i \int fg_2\, d|m|$, where $g_1$ and $g_2$ belong to $\mathscr{L}^q$ (Ch. V, §5, No. 8, Th. 4); whence the final assertion, on setting $g = (g_1 + ig_2)h^{-1}$.