Hölder inequality

The Hölder inequalityMathworldPlanetmath concerns vector p-norms: given 1p, q,

If 1p+1q=1 then |xTy|||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 Lp spaces (http://planetmath.org/LpSpace). If a function f is in Lp(X), then the Lp-norm of f is denoted ||f||p. Given a measure spaceMathworldPlanetmath (X,𝔅,μ), if f is in Lp(X) and g is in Lq(X) (with 1/p+1/q=1), then the Hölder inequality becomes

fg1=X|fg|dμ (X|f|pdμ)1p(X|g|qdμ)1q
= fpgq
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 inequalityMathworldPlanetmath
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