Let \((\Omega, \Sigma, \mu)\) be a meausurable space and let \(p,q\in [1,\infty]\) with \(\frac{1}{p}+\frac{1}{q}=1\). Then
\begin{equation*} \lVert fg\rVert_1 \le \lVert f\rVert_p \lVert g\rVert_q \end{equation*}for all \(f\in L^p(\mu)\) and \(g\in L^q(\mu)\).