---
tag: 0BKDAGMGX9
kind: statement
role: theorem
name: Hölder's inequality for sums
msc: [26D15]
uses: [07VHQ31EYJ]
sources:
  - url: https://en.wikipedia.org/wiki/Hölder%27s_inequality
    locator: Counting measure
---

## Statement

Let $p, q > 1$ be real numbers with $\frac{1}{p} + \frac{1}{q} = 1$, and let $a_1, \dots, a_n, b_1, \dots, b_n$ be real numbers.
Then
$$\sum_{i=1}^{n} |a_i b_i| \le \Big( \sum_{i=1}^{n} |a_i|^p \Big)^{1/p} \Big( \sum_{i=1}^{n} |b_i|^q \Big)^{1/q}.$$

## Remarks

Scale so that both factors on the right are $1$. Then apply [[Young's inequality|07VHQ31EYJ]] to each product $|a_i| |b_i|$ and add.
For $p = q = 2$ it is the Cauchy–Schwarz inequality.
Mathlib states a form as `Real.inner_le_Lp_mul_Lq`.

## Explanations

### L1 human

Hölder's inequality bounds a sum of products by the sizes of the two sequences.
Each size is measured with its own exponent, $p$ for one and $q$ for the other.