https://wikimd.org/index.php?action=history&feed=atom&title=Lp_spaceLp space - Revision history2026-04-25T15:48:13ZRevision history for this page on the wikiMediaWiki 1.44.2https://wikimd.org/index.php?title=Lp_space&diff=5611739&oldid=prevPrab: CSV import2024-04-16T06:34:48Z<p>CSV import</p>
<p><b>New page</b></p><div>[[Image:Vector-p-Norms_qtl1.svg|left|Vector-p-Norms qtl1|thumb]] [[Image:Astroid.svg|left|Astroid|thumb|left]] [[File:Lp_space_animation.gif|left|Lp space animation|thumb]] '''Lp spaces''' are a family of [[functional analysis|functional spaces]] that are pivotal in various branches of [[mathematics]], including [[analysis]], [[probability theory]], and [[statistics]]. They are defined using a generalization of the [[Pythagorean theorem]] and are essential in the study of [[Lebesgue integration]], [[Fourier analysis]], and many other areas of pure and applied mathematics.<br />
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== Definition ==<br />
An ''Lp space'' is a vector space of functions for which the ''p''-th power of the absolute value of the function is [[Lebesgue integration|integrable]]. Formally, given a measure space \((X, \Sigma, \mu)\), the Lp space \(L^p(X, \Sigma, \mu)\) consists of all [[measurable function|measurable functions]] \(f\) for which the [[norm (mathematics)|norm]] defined by<br />
<br />
\[<br />
\|f\|_p = \left( \int_X |f|^p \, d\mu \right)^{1/p}<br />
\]<br />
<br />
is finite. Here, \(1 \leq p < \infty\). For \(p = \infty\), the Lp space \(L^\infty\) is defined using the [[essential supremum]]:<br />
<br />
\[<br />
\|f\|_\infty = \inf\{C \geq 0 : \mu(\{x \in X : |f(x)| > C\}) = 0\}.<br />
\]<br />
<br />
== Properties ==<br />
Lp spaces have several important properties that make them useful in analysis and applied mathematics:<br />
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- '''Completeness''': Every Lp space is a [[complete space]], meaning that every [[Cauchy sequence]] in \(L^p\) converges to an element in \(L^p\). This property classifies Lp spaces as [[Banach spaces]].<br />
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- '''Separability''': For \(1 \leq p < \infty\), \(L^p\) spaces are [[separable space|separable]], meaning they contain a countable, dense subset. This property is crucial for the application of various analytical techniques.<br />
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- '''Convexity''': Lp spaces are [[convex set|convex]], which has implications for optimization and functional analysis.<br />
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- '''Duality''': For \(1 < p < \infty\), the dual space of \(L^p\) is \(L^q\), where \(\frac{1}{p} + \frac{1}{q} = 1\). This relationship is fundamental in the study of [[functional analysis]] and has numerous applications.<br />
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== Applications ==<br />
Lp spaces are used in a wide range of mathematical and applied contexts:<br />
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- In [[Fourier analysis]], Lp spaces provide a framework for understanding the convergence of [[Fourier series]] and [[Fourier transform]]s.<br />
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- In [[partial differential equations]], solutions and their properties are often studied within the context of Lp spaces.<br />
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- In [[probability theory]], Lp spaces are used to define and analyze [[random variables]] and [[expectation values]], particularly in the context of [[Lp spaces on probability spaces]].<br />
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- In [[numerical analysis]] and [[approximation theory]], Lp norms are used to measure the error between a function and its approximation.<br />
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== See Also ==<br />
- [[Banach space]]<br />
- [[Hilbert space]]<br />
- [[Norm (mathematics)]]<br />
- [[Lebesgue integration]]<br />
- [[Measure (mathematics)]]<br />
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[[Category:Functional analysis]]<br />
[[Category:Mathematical analysis]]<br />
[[Category:Measure theory]]<br />
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