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![[parent]](http://images.planetmath.org:8080/images/uparrow.png)
 -norm is dual to 
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(Theorem)
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If $(X,\mathfrak{M},\mu)$ is any measure space and $1\le p,q\le \infty$ are Hölder conjugates then, for $f\in L^p$ , the following linear function can be defined
The Hölder inequality shows that this gives a well defined and bounded linear map. Its operator norm is given by \begin{equation*} \Vert\Phi_f\Vert=\left\{\Vert fg\Vert_1:g\in L^q, \Vert g\Vert_q=1\right\}. \end{equation*}The following theorem shows that the operator norm of $\Phi_f$ is equal to the $L^p$ -norm of $f$ .
Theorem Let $(X,\mathfrak{M},\mu)$ be a $\sigma$ -finite measure space and $p,q$ be Hölder conjugates. Then, any measurable function $f\colon X\rightarrow\mathbb{C}$ has $L^p$ -norm \begin{equation}\label{eq:1} \Vert f\Vert_p=\sup\left\{\Vert fg\Vert_1: g\in L^q, \Vert g\Vert_q=1\right\}. \end{equation}Furthermore, if either $p<\infty$ and $\Vert f\Vert_p<\infty$ or $p=1$ then $\mu$ is not required to be $\sigma$ -finite.
Note that the $\sigma$ -finite condition is required, except in the cases mentioned. For example, if $\mu$ is the measure satisfying $\mu(A)=\infty$ for every nonempty set $A$ , then $L^p(\mu)=\{0\}$ for $p<\infty$ and it is easily checked that equality (![[*]](http://images.planetmath.org:8080/cache/objects/11377/js//usr/share/latex2html/icons/crossref.png "[*]") ) fails whenever $f=1$ and $p>1$ .
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Cross-references: equality, measure, measurable function, conjugates, theorem, operator norm, bounded linear map, well defined, function, measure space
There are 2 references to this entry.
This is version 2 of  -norm is dual to  , born on 2008-12-23, modified 2008-12-24.
Object id is 11377, canonical name is LpNormIsDualToLq.
Accessed 361 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) | | | 28A25 (Measure and integration :: Classical measure theory :: Integration with respect to measures and other set functions) |
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Pending Errata and Addenda
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