Hölder's inequality for sums

Builds on Young's inequality for products

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 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.

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Inequality of arithmetic and geometric means statement
Young's inequality for products statement
Hölder's inequality for sums statement

Tag	`0BKDAGMGX9`
URI	`m2://0BKDAGMGX9`
Kind	statement, theorem
Writing	W1 stated
MSC	26D15
Status	This copy of the site has no build data. Run `m2 status 0BKDAGMGX9`.
Uses	Young's inequality for products

Hölder's inequality for sums

Statement

Remarks

Reading path

Further reading

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