# Hölder inequality

The Hölder inequality concerns vector p-norms: given $1\leq p$, $q\leq\infty$,

 $\mbox{If }\frac{1}{p}+\frac{1}{q}=1\mbox{ then }|x^{T}y|\leq||\,x\,||_{p}||\,y% \,||_{q}$

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 $(X,\mathfrak{B},\mu)$, 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

 $\displaystyle\|fg\|_{1}=\int_{X}|fg|\mathrm{d}\mu$ $\displaystyle\leq$ $\displaystyle\left(\int_{X}|f|^{p}\mathrm{d}\mu\right)^{\frac{1}{p}}\left(\int% _{X}|g|^{q}\mathrm{d}\mu\right)^{\frac{1}{q}}$ $\displaystyle=$ $\displaystyle\|f\|_{p}\,\|g\|_{q}$
 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