Fréchet–Kolmogorov theorem

In functional analysis, the Fréchet–Kolmogorov theorem (the names of Riesz or 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. It can be thought of as an Lp version of the Arzelà–Ascoli theorem, from which it can be deduced. The theorem is named after Maurice René Fréchet and Andrey Kolmogorov.

Statement

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Let   be a subset of   with  , and let   denote the translation of   by  , that is,  

The subset   is relatively compact if and only if the following properties hold:

  1. (Equicontinuous)   uniformly on  .
  2. (Equitight)   uniformly on  .

The first property can be stated as   such that   with  

Usually, the Fréchet–Kolmogorov theorem is formulated with the extra assumption that   is bounded (i.e.,   uniformly on  ). However, it has been shown that equitightness and equicontinuity imply this property.[1]

Special case

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For a subset   of  , where   is a bounded subset of  , the condition of equitightness is not needed. Hence, a necessary and sufficient condition for   to be relatively compact is that the property of equicontinuity holds. However, this property must be interpreted with care as the below example shows.

Examples

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Existence of solutions of a PDE

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Let   be a sequence of solutions of the viscous Burgers equation posed in  :

 

with   smooth enough. If the solutions   enjoy the  -contraction and  -bound properties,[2] we will show existence of solutions of the inviscid Burgers equation

 

The first property can be stated as follows: If   are solutions of the Burgers equation with   as initial data, then

 

The second property simply means that  .

Now, let   be any compact set, and define

 

where   is   on the set   and 0 otherwise. Automatically,   since

 

Equicontinuity is a consequence of the  -contraction since   is a solution of the Burgers equation with   as initial data and since the  -bound holds: We have that

 

We continue by considering

 

The first term on the right-hand side satisfies

 

by a change of variable and the  -contraction. The second term satisfies

 

by a change of variable and the  -bound. Moreover,

 

Both terms can be estimated as before when noticing that the time equicontinuity follows again by the  -contraction.[3] The continuity of the translation mapping in   then gives equicontinuity uniformly on  .

Equitightness holds by definition of   by taking   big enough.

Hence,   is relatively compact in  , and then there is a convergent subsequence of   in  . By a covering argument, the last convergence is in  .

To conclude existence, it remains to check that the limit function, as  , of a subsequence of   satisfies

 

See also

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References

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  1. ^ Sudakov, V.N. (1957). "Criteria of compactness in function spaces". (In Russian), Upsekhi Math. Nauk. 12: 221–224. {{cite journal}}: Cite journal requires |journal= (help)
  2. ^ Necas, J.; Malek, J.; Rokyta, M.; Ruzicka, M. (1996). Weak and Measure-Valued Solutions to Evolutionary PDEs. Applied Mathematics and Mathematical Computation 13. Chapman and Hall/CRC. ISBN 978-0412577505.
  3. ^ Kruzhkov, S. N. (1970). "First order quasi-linear equations in several independent variables". Math. USSR Sbornik. 10 (2): 217–243. doi:10.1070/SM1970v010n02ABEH002156.

Literature

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