proof of Hölder inequality

First we prove the more general form (in measure spaces).

Let $(X,\mu )$ be a measure space and let $f\in L^p(X)$, $g\in L^q(X)$ where $p,q\in [1,+\mathrm{\infty }]$ and $\frac{1}{p}+\frac{1}{q}=1$.

The case $p=1$ and $q=\mathrm{\infty }$ is obvious since

$$|f(x)g(x)|\le {\parallel g\parallel }_{L^{\mathrm{\infty }}}|f(x)|.$$

Also if $f=0$ or $g=0$ the result is obvious. Otherwise notice that (applying http://planetmath.org/node/YoungInequalityYoung inequality) we have

$$\frac{{\parallel fg\parallel }_1}{{\parallel f\parallel }_p\cdot {\parallel g\parallel }_q}={\int }_X\frac{|f|}{{\parallel f\parallel }_p}\cdot \frac{|g|}{{\parallel g\parallel }_q}𝑑\mu \le \frac{1}{p}{\int }_X{\left(\frac{|f|}{{\parallel f\parallel }_p}\right)}^p𝑑\mu +\frac{1}{q}{\int }_X{\left(\frac{|g|}{{\parallel g\parallel }_q}\right)}^q𝑑\mu =\frac{1}{p}+\frac{1}{q}=1$$

hence the desired inequality holds

$${\int }_X|fg|={\parallel fg\parallel }_1\le {\parallel f\parallel }_p\cdot {\parallel g\parallel }_q={\left({\int }_X{|f|}^p\right)}^{\frac{1}{p}}{\left({\int }_X{|g|}^q\right)}^{\frac{1}{q}}.$$

If $x$ and $y$ are vectors in ${ℝ}^n$ or vectors in ${\mathrm{\ell }}^p$ and ${\mathrm{\ell }}^q$-spaces we can specialize the previous result by choosing $\mu$ to be the counting measure on $\mathbb{N}$.

In this case the proof can also be rewritten, without using measure theory, as follows. If we define

$${\parallel x\parallel }_p={\left(\sum _k{|x_k|}^p\right)}^{\frac{1}{p}}$$

we have

$$\frac{\left|{\sum }_k{x}_k{y}_k\right|}{{\parallel x\parallel }_p\cdot {\parallel y\parallel }_q}\le \frac{{\sum }_k|x_k||y_k|}{{\parallel x\parallel }_p\cdot {\parallel y\parallel }_q}=\sum _k\frac{|x_k|}{{\parallel x\parallel }_p}\frac{|y_k|}{{\parallel y\parallel }_q}\le \frac{1}{p}\sum _k\frac{{|x_k|}^p}{{\parallel x\parallel }_p^p}+\frac{1}{q}\sum _k\frac{{|y_k|}^q}{{\parallel y\parallel }_q^q}=\frac{1}{p}+\frac{1}{q}=1.$$

Title	proof of Hölder inequality
Canonical name	ProofOfHolderInequality
Date of creation	2013-03-22 13:31:16
Last modified on	2013-03-22 13:31:16
Owner	paolini (1187)
Last modified by	paolini (1187)
Numerical id	10
Author	paolini (1187)
Entry type	Proof
Classification	msc 15A60
Classification	msc 46E30
Synonym	proof of Hölder inequality
Synonym	proof of Holder’s inequality