Category:Hölder's Inequality for Integrals
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This category contains pages concerning Hölder's Inequality for Integrals:
Let $\struct {X, \Sigma, \mu}$ be a measure space.
Let $p, q \in \R_{>0}$ such that $\dfrac 1 p + \dfrac 1 q = 1$.
 | Although this article appears correct, it's inelegant. There has to be a better way of doing it. In particular: the assumption should read $p,q\in\R_{>0}\cup\set{+\infty}$. Suggestion: make a page for defining $p,q$ as satisfying this relation, including the pair $\tuple{1,\infty}$ You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by redesigning 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 {{Improve}} from the code.If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page. |
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 their pointwise product $f g$ is $\mu$-integrable, that is:
- $f g \in \map {\LL^1} \mu$
and:
| \(\ds \norm {f g}_1\) | \(=\) | \(\ds \int \size {f g} \rd \mu\) | ||||||||||||
| \(\ds \) | \(\le\) | \(\ds \paren {\int \size f^p \rd \mu}^{1 / p} \paren {\int \size g^q \rd \mu}^{1 / q}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \norm f_p \cdot \norm g_q\) |
where:
- $\size {f g}$ denotes the absolute value function applied to the pointwise product of $f$ and $g$
- the $\norm {\, \cdot \,}_p$ signify $p$-seminorms.
Source of Name
This entry was named for Otto Ludwig Hölder.
Pages in category "Hölder's Inequality for Integrals"
The following 4 pages are in this category, out of 4 total.