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Hölder inequality
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(Theorem)
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The Hölder inequality concerns vector p-norms: given $1 \leq p$ , $q \leq \infty$ ,
If  then  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
\begin{eqnarray*} \Vert fg\Vert_1 = \int_X \vert fg\vert \mathrm{d}\mu & \le & \left(\int_X|f|^p\mathrm{d}\mu\right)^{\frac{1}{p}} \left(\int_X|g|^q\mathrm{d}\mu\right)^{\frac{1}{q}}\\ & = & \Vert f\Vert_p\,\Vert g \Vert_q \end{eqnarray*}
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"Hölder inequality" is owned by PrimeFan. [ full author list (6) | owner history (5) ]
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Cross-references: measure space, function, Cauchy-Schwarz inequality, vector p-norms
There are 14 references to this entry.
This is version 15 of Hölder inequality, born on 2001-10-06, modified 2008-10-04.
Object id is 94, canonical name is HolderInequality.
Accessed 36117 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) |
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Pending Errata and Addenda
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