Let $(X,\mathcal{A},\mu)$ be a [measure space](/page/Measure%20Space), let $\mathbb{F}\in\{\mathbb{R},\mathbb{C}\}$, let $1 \leq p \leq \infty$, and let $q$ be the conjugate exponent to $p$, so that $q=\infty$ when $p=1$, $q=1$ when $p=\infty$, and $\frac{1}{p}+\frac{1}{q}=1$ when $1<p<\infty$. For every $g \in L^q(X,\mathcal{A},\mu;\mathbb{F})$, define

\begin{align*} \Lambda_g: L^p(X,\mathcal{A},\mu;\mathbb{F}) \to \mathbb{F}, \quad f \mapsto \int_X f\overline{g}\,d\mu(x). \end{align*}

Here complex conjugation is interpreted as the identity map when $\mathbb{F}=\mathbb{R}$. Then $\Lambda_g$ is a bounded linear functional on $L^p(X,\mathcal{A},\mu;\mathbb{F})$, and

\begin{align*} \|\Lambda_g\|_{(L^p)^*} \leq \|g\|_{L^q}. \end{align*}