Hölder inequality (integrals)

Let $(X,\Sigma,\mu)$ be a measure space and let $1\le p,q\le\infty$ satisfy $1/p+1/q=1$ , with $1/\infty=0$ . For real or complex $f\in L^p$ and $g\in L^q$ , Hölder's inequality is

\int_X|fg|\,d\mu\le\|f\|_p\|g\|_q.

Proof and endpoints

For $1<p,q<\infty$ and nonzero norms, normalize both functions to norm one, apply Young's product inequality pointwise, and integrate. If a norm is zero, the corresponding function vanishes almost everywhere. At the endpoint $p=1,q=\infty$ , use $|fg|\le|f|\|g\|_\infty$ almost everywhere; the other endpoint is symmetric.

Special cases

Counting measure gives the finite-sum and sequence versions. For $p=q=2$ , this is the integral Cauchy–Schwarz inequality. On a measurable set $E$ of finite measure, applying it to $|f|$ and $\mathbf1_E$ gives $\|f\|_{L^1(E)}\le\mu(E)^{1-1/p}\|f\|_{L^p(E)}$ .

References

John K. Hunter, Measure Theory, Theorem 7.7.