Hölder's Inequality for Integrals/Equality

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Theorem

Let $\struct {X, \Sigma, \mu}$ be a measure space.

Let $p, q \in \R_{>0}$ such that $\dfrac 1 p + \dfrac 1 q = 1$.

Let $f \in \map {\LL^p} \mu, f: X \to \R$, and $g \in \map {\LL^q} \mu, g: X \to \R$, where $\LL$ denotes Lebesgue space.


Then equality in Hölder's Inequality for Integrals, that is:

$\ds \int \size {f g} \rd \mu = \norm f_p \cdot \norm g_q$

holds if and only if, for $\mu$-almost all $x \in X$:

$\dfrac {\size {\map f x}^p} {\norm f_p^p} = \dfrac {\size {\map g x}^q} {\norm g_q^q}$


The validity of the material on this page is questionable.
In particular: What if e.g. $f=0$ and $g=1$? The equality holds, trivially, but how to read the necessary condition?
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This page needs the help of a knowledgeable authority.
In particular: Whoever is on the case, note that the equality condition as given in both Abramowitz & Stegun and Spiegel is as follows, but then they don't go into that fancy-pants $\norm f_p$ notation either, they specifically use the notation $\ds \paren {\int_a^b {\size {\map f x} }^p \rd x}^{1 / p}$ which is more accessible, although admittedly in the context of a Riemann integral rather than a Lebesgue one. Input needed from someone who a) is familiar with the theorem in the general Lebesgue space and b) is motivated to bring the exposition back down to earth by explaining what $\norm f_p$ actually means in undergrad language.
If you are knowledgeable in this area, then you can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by resolving the issues.
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When this work has been completed, you may remove this instance of {{Help}} from the code.
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$\dfrac {\size {\map f x}^{p - 1} } {\size {\map g x} } = c$

for some $c \in \R$.



This article, or a section of it, needs explaining.
In particular: The notation $\norm f_p^p$ etc. is opaque and confusing. It probably means $\paren {\norm f_p}^p$ where ${}^p$ denotes power, and we could probably better express it as ${\norm f_p}^p$.
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it.
To discuss this page in more detail, feel free to use the talk page.
When this work has been completed, you may remove this instance of {{Explain}} from the code.


Proof


This theorem requires a proof.
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof.
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When this work has been completed, you may remove this instance of {{ProofWanted}} from the code.
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Source of Name

This entry was named for Otto Ludwig Hölder.


Sources

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