= Hölder's inequality
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{title2=$\lVert fg\rVert_1\leq\lVert f\rVert_p\lVert g\rVert_q$}
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<Hölder's inequality> controls a product using separate information about its two factors. If one factor is large on part of a domain, the other factor's complementary <Lp norm> limits how much their product can accumulate there. The same statement bounds finite <sums>, <integrals> and <expected values>; no <independence> is required in the probabilistic version.
Let $p,q\in[1,\infty]$ be <conjugate exponents>, meaning
$$
\frac1p+\frac1q=1,\qquad\frac1\infty=0.
$$
On a <measure space>, if $f\in L^p$ and $g\in L^q$, then $fg\in L^1$ and
$$
\boxed{\int |fg|\,d\mu\leq\lVert f\rVert_p\lVert g\rVert_q.}
$$
Here the <Lp norms> are finite; at exponent $\infty$ the norm uses the <essential supremum>. For a finite <vector>, integration with <counting measure> gives
$$
\left|\sum_i x_i y_i\right|\leq\sum_i|x_i y_i|
\leq\left(\sum_i|x_i|^p\right)^{1/p}
\left(\sum_i|y_i|^q\right)^{1/q},\qquad 1<p,q<\infty.
$$
The first step is the <triangle inequality>; the second is <Hölder's inequality>.<NIST Digital Library of Mathematical Functions/1 Algebraic and Analytic Methods>[NIST Digital Library of Mathematical Functions, §1.7(i)–(ii), equations 1.7.2 and 1.7.5 and the accompanying equality statements]{ref}
For finite <vectors>, the endpoint case $p=1$, $q=\infty$ is $|x^Ty|\leq\lVert x\rVert_1\lVert y\rVert_\infty$: bound every $|y_i|$ by its <maximum>, then sum the $|x_i|$. Swapping the factors gives the other endpoint. The choice $p=q=2$ gives the <Cauchy-Schwarz inequality>.
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