Hölder inequality (finite sums)

Proposition (Hölder inequality for finite sums). Let $x_i,y_i\in\mathbb{R}$ for $i=1,\dots,m$ . If $p>1$ and $q>1$ satisfy $1/p+1/q=1$ , then

\sum_{i=1}^m |x_i y_i| \le \left(\sum_{i=1}^m |x_i|^p\right)^{1/p} \left(\sum_{i=1}^m |y_i|^q\right)^{1/q}.

Remarks

Context. This is the fundamental inequality behind duality of $\ell^p$ spaces and many estimates in analysis. It can be proved using the weighted AM–GM inequality (or Young's inequality).

Proof sketch. Normalize so that $\sum |x_i|^p=1$ and $\sum |y_i|^q=1$ (otherwise scale). Apply weighted AM–GM with $\theta=1/p$ to $a=|x_i|^p$ and $b=|y_i|^q$ to get

|x_i y_i|\le \frac{|x_i|^p}{p}+\frac{|y_i|^q}{q}.

Sum over $i$ and undo the normalization.