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 \(g\in L^q(\mu)\).
Proof
The inequality holds for \(p=1\) and \(q=\infty\) or vice versa, and for \(f\equiv 0\) or \(g\equiv 0\). Assume \(p,q\in (1,\infty )\). Using Young’s inequality for products we find for \(\omega\in \Omega\)
\[ \frac{\lvert f(\omega)\rvert\lvert g(\omega)\rvert}{\lVert f\rVert_p \lVert g\rVert_q}\le \frac{\lvert f(\omega)\rvert^p}{p\lVert f\rVert^p_p}+\frac{\lvert g(\omega)\rvert^q}{q\lVert g\rVert^q_q}. \]Integrating over both sides yields Hölder’s inequality.