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bounded linear functionals on 
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If $\mu$ is a positive measure on a set $X$ , $1\leq p\leq\infty$ , and $g\in L^q(\mu)$ , where $q$ is the Hölder conjugate of $p$ , then Hölder's inequality implies that the map $f\mapsto\int_Xfgd\mu$ is a bounded
linear functional on $L^p(\mu)$ . It is therefore natural to ask whether or not all such functionals on $L^p(\mu)$ are of this form for some $g\in L^q(\mu)$ . Under fairly mild hypotheses, and excepting the case $p=\infty$ , the Radon-Nikodym Theorem answers this question affirmatively.
Theorem Let $(X,\mathfrak{M},\mu)$ be a $\sigma$ -finite measure space, $1\leq p<\infty$ , and $q$ the Hölder conjugate of $p$ . If $\Phi$ is a bounded linear functional on $L^p(\mu)$ , then there exists a unique $g\in L^q(\mu)$ such that \begin{equation} \Phi(f)=\int_Xfgd\mu \end{equation}for all $f\in L^p(\mu)$ . Furthermore, $\norm{\Phi}=\norm{g}_q$ . Thus, under the stated hypotheses, $L^q(\mu)$ is isometrically isomorphic to the dual space of $L^p(\mu)$ .
If $1<p<\infty$ , then the assertion of the theorem remains valid without the assumption that $\mu$ is $\sigma$ -finite; however, even with this hypothesis, the result can fail in the case that $p=\infty$ . In particular, the bounded linear functionals on $L^\infty(m)$ , where $m$ is Lebesgue measure on $[0,1]$
, are not all obtained in the above manner via members of $L^1(m)$ . An explicit example illustrating this is constructed as follows: the assignment $f\mapsto f(0)$ defines a bounded linear functional on $C([0,1])$ , which, by the Hahn-Banach Theorem, may be extended to a bounded linear functional $\Phi$ on $L^\infty(m)$ . Assume for the sake of contradiction that there exists $g\in L^1(m)$ such that $\Phi(f)=\int_{[0,1]}fgdm$ for every $f\in L^\infty(m)$ , and for $n\in\mathbb{Z}^+$ , define $f_n:[0,1]\rightarrow\mathbb{C}$ by
$f_n(x)=\max\set{1-nx,0}$ . As each $f_n$ is continuous, we have $\Phi(f_n)=\varphi(f_n)=1$ for all $n$ ; however, because $f_n\rightarrow 0$ almost everywhere and $\abs{f_n}\leq 1$ , the Dominated Convergence Theorem, together with our hypothesis on $g$ , gives \begin{equation*} 1=\lim_{n\rightarrow\infty}\Phi(f_n)=\lim_{n\rightarrow\infty}\int_{[0,1]}f_ngdm=0\text{,} \end{equation*}a contradiction. It follows that no such $g$ can exist.
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See Also:  -space, Hölder inequality, continuous linear mapping, Banach space, dual space, conjugate index, Radon-Nikodym theorem, bounded linear functionals on  , -norm is dual to 
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linear functional, dual space, conjugate exponent |
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Cross-references: dominated convergence theorem, almost everywhere, continuous, contradiction, Hahn-Banach theorem, Lebesgue measure, hypothesis, even, valid, theorem, dual space, isometrically isomorphic, measure space, Radon-Nikodym theorem, functionals, linear functional, bounded, map, implies, Hölder's inequality, conjugate, positive measure
This is version 12 of bounded linear functionals on  , born on 2008-11-22, modified 2008-12-21.
Object id is 11269, canonical name is BoundedLinearFunctionalsOnLpmu.
Accessed 892 times total.
Classification:
| AMS MSC: | 28B15 (Measure and integration :: Set functions, measures and integrals with values in abstract spaces :: Set functions, measures and integrals with values in ordered spaces) |
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Pending Errata and Addenda
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