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Hölder inequality (Theorem)

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

   If $\displaystyle \frac{1}{p}+\frac{1}{q}=1$ then $\displaystyle \vert x^Ty\vert \leq \vert\vert\,x\,\vert\vert _p\vert\vert\,y\,\vert\vert _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 $ \vert\vert\,f\,\vert\vert _p$. Let $ (X,\mathfrak{B},\mu)$ be a measure space. 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


$\displaystyle \Vert fg\Vert_1 = \int_X \vert fg\vert \mathrm{d}\mu$ $\displaystyle \le$ $\displaystyle \left(\int_X\vert f\vert^p\mathrm{d}\mu\right)^{\frac{1}{p}} \left(\int_X\vert g\vert^q\mathrm{d}\mu\right)^{\frac{1}{q}}$  
  $\displaystyle =$ $\displaystyle \Vert f\Vert_p\,\Vert g \Vert_q$  



"Hölder inequality" is owned by Mravinci. [ full author list (5) | owner history (4) ]
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See Also: vector p-norm, Cauchy-Schwartz inequality, Cauchy-Schwarz inequality, proof of Minkowski inequality, conjugate index

Other names:  Holder inequality, Hoelder inequality
Keywords:  vector, norm

Pronunciation (guide):
 Holder: /hul-dr/

Attachments:
proof of Hölder inequality (Proof) by paolini
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Cross-references: measure space, function, Cauchy-Schwarz inequality, vector p-norms
There are 10 references to this entry.

This is version 14 of Hölder inequality, born on 2001-10-06, modified 2006-09-09.
Object id is 94, canonical name is HolderInequality.
Accessed 31565 times total.

Classification:
AMS MSC: 46E30 (Functional analysis :: Linear function spaces and their duals :: Spaces of measurable functions ($L^p$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant)
 15A60 (Linear and multilinear algebra; matrix theory :: Norms of matrices, numerical range, applications of functional analysis to matrix theory)
Pending Errata and Addenda
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Discussion
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More that just "vector" p-norms by ariels on 2002-06-03 07:56:32
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.
[ reply | up ]
layout? by drini on 2001-10-06 03:30:39
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 
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