Hölder inequality

The Hölder inequality concerns vector p-norms: given $1\le p$, $q\le \mathrm{\infty }$,

$$\text{If }\frac{1}{p}+\frac{1}{q}=1\text{ then }|x^{T}y|\le {||x||}_p{||y||}_q$$

An important instance of a Hölder inequality is the Cauchy-Schwarz inequality.

There is a version of this result for the $L^p$ spaces (http://planetmath.org/LpSpace). If a function $f$ is in $L^p(X)$, then the $L^p$-norm of $f$ is denoted ${||f||}_p$. Given a measure space $(X,𝔅,\mu )$, if $f$ is in $L^p(X)$ and $g$ is in $L^q(X)$ (with $1/p+1/q=1$), then the Hölder inequality becomes

${\parallel fg\parallel }_1={\displaystyle {\int }_X}|fg|d\mu$ $\le$ ${\left({\displaystyle {\int }_X}{|f|}^{p}d\mu \right)}^{\frac{1}{p}}{\left({\displaystyle {\int }_X}{|g|}^{q}d\mu \right)}^{\frac{1}{q}}$

$=$ ${\parallel f\parallel }_p{\parallel g\parallel }_q$

Title	Hölder inequality
Canonical name	HolderInequality
Date of creation	2013-03-22 11:43:06
Last modified on	2013-03-22 11:43:06
Owner	PrimeFan (13766)
Last modified by	PrimeFan (13766)
Numerical id	27
Author	PrimeFan (13766)
Entry type	Theorem
Classification	msc 15A60
Classification	msc 55-XX
Classification	msc 46E30
Classification	msc 42B10
Classification	msc 42B05
Synonym	Holder inequality
Synonym	Hoelder inequality
Related topic	VectorPnorm
Related topic	CauchySchwartzInequality
Related topic	CauchySchwarzInequality
Related topic	ProofOfMinkowskiInequality
Related topic	ConjugateIndex
Related topic	BoundedLinearFunctionalsOnLpmu
Related topic	ConvolutionsOfComplexFunctionsOnLocallyCompactGroups
Related topic	LpNormIsDualToLq