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| In [[functional analysis]], the [[Maurice Fréchet|Fréchet]]-[[Andrey Kolmogorov|Kolmogorov]] theorem (the names of [[Marcel Riesz|Riesz]] or [[André Weil|Weil]] are sometimes added as well) gives a necessary and sufficient condition for a set of functions to be [[relatively compact]] in an [[Lp space|L<sup>p</sup> space]]. It can be thought of as an L<sup>p</sup> version of the [[Arzelà-Ascoli theorem]], from which it can be deduced.
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| == Statement ==
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| Let <math>B</math> be a bounded set in <math>L^p(\mathbb{R}^n)</math>, with <math>p\in[1,\infty)</math>.
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| The subset ''B'' is [[Relatively compact subspace|relatively compact]] if and only if the following properties hold:
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| #<math>\lim_{r\to\infty}\int_{|x|>r}\left|f\right|^p=0</math> uniformly on ''B'',
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| #<math>\lim_{a\to 0}\Vert\tau_a f-f\Vert_{L^p(\mathbb{R}^n)} = 0</math> uniformly on ''B'',
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| where <math>\tau_a f</math> denotes the translation of <math>f</math> by <math>a</math>, that is, <math>\tau_a f(x)=f(x-a) .</math>
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| The second property can be stated as <math>\forall \varepsilon >0 \, \, \exists \delta >0 </math> such that <math>\Vert\tau_a f-f\Vert_{L^p(\mathbb{R}^n)} < \varepsilon \, \, \forall f \in B, \forall a</math> with <math>|a|<\delta .</math>
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| == References ==
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| * {{cite book
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| | last = Brezis
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| | first = Haïm
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| | authorlink = Haïm Brezis
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| | title = Functional analysis, Sobolev spaces, and partial differential equations
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| | publisher = [[Springer-Verlag|Springer]]
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| | series = Universitext
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| | year = 2010
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| | isbn = 978-0-387-70913-0
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| | page = 111
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| }}
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| * [[Marcel Riesz]], « [http://acta.fyx.hu/acta/showCustomerArticle.action?id=5397&dataObjectType=article Sur les ensembles compacts de fonctions sommables] », dans ''[[Acta Scientiarum Mathematicarum|Acta Sci. Math.]]'', vol. 6, 1933, p. 136–142
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| * {{cite book
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| | last = Precup
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| | first = Radu
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| | title = Methods in nonlinear integral equations
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| | publisher = [[Springer-Verlag|Springer]]
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| | year = 2002
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| | isbn = 978-1-4020-0844-3
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| | page = 21
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| }}
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| {{DEFAULTSORT:Frechet-Kolmogorov Theorem}}
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| [[Category:Theorems in functional analysis]]
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| [[Category:Compactness theorems]]
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| {{Mathanalysis-stub}}
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6 They call the author Kelly Lyman and thinks it sounds quite very good. To read comics is an issue that I'm totally addicted on to. Dispatching is my profession. New York is his birth place along with the family loves it. If you want to be aware of more the look at my website: http://onlog.com.br/ajuda/index.php?title=How_you_Ought_To_Hire_A_Dj_For_Your_Wedding
my website; awesome home Theater (onlog.com.Br)