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# 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. 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

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

$\displaystyle=$ | $\displaystyle\|f\|_{p}\,\|g\|_{q}$ |

Keywords:

vector, norm

Related:

VectorPnorm, CauchySchwartzInequality, CauchySchwarzInequality, ProofOfMinkowskiInequality, ConjugateIndex, BoundedLinearFunctionalsOnLpmu, ConvolutionsOfComplexFunctionsOnLocallyCompactGroups, LpNormIsDualToLq

Synonym:

Holder inequality, Hoelder inequality

Type of Math Object:

Theorem

Major Section:

Reference

Groups audience:

## Mathematics Subject Classification

15A60*no label found*55-XX

*no label found*46E30

*no label found*42B10

*no label found*42B05

*no label found*

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## Info

## Attached Articles

## Corrections

positive p and q by AxelBoldt ✘

this is not a definition! by paolini ✓

Other names by yark ✓

clarify by Mathprof ✓

holder conjugates by gel ✓

this is not a definition! by paolini ✓

Other names by yark ✓

clarify by Mathprof ✓

holder conjugates by gel ✓

## Comments

## layout?

perhaps

if $p$ and $q$ are such that $1/p+1/q=1$ then...

cause it looks like you're multiplying the norms with the fractions...

f

G -----> H G

p \ /_ ----- ~ f(G)

\ / f ker f

G/ker f

## More that just "vector" p-norms

The H\"older inequality applies generally to objects in Banach spaces (also infinite dimensional) $L_p(X)$ and $L_q(X)$ (where, as always, $\frac{1}{p}+\frac{1}{q}=1$); it states that the product is integrable (a member of $L_1(X)$), and that the norms behave as required.