H\"older inequalityHolderInequality2001-10-06 03:16:322008-10-04 19:45:51Theorem/hul-dr/vectornorm\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}The \emph{H\"older inequality} concerns \emph{vector p-norms}: given $1 \leq p$, $q \leq \infty$,
\begin{displaymath}
\mbox{If }\frac{1}{p}+\frac{1}{q}=1\mbox{ then }|x^Ty| \leq ||\,x\,||_p||\,y\,||_q
\end{displaymath}
An important instance of a H\"older inequality is the \emph{Cauchy-Schwarz inequality}.
There is a version of this result for the \PMlinkname{$L^p$ spaces}{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 H\"older 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*}