Fréchet-Kolmogorov theorem

The set of Kolmogorov - Riesz (after Andrei Nikolaevich Kolmogorov and Marcel Riesz ) is a theorem from the mathematical branch of functional analysis, which is a compactness criterion for subsets of Lp- spaces. This rate is, depending on degree of generalization, also set of M. Riesz, set of Kolmogorov - Fréchet - Riesz or set of Kolmogorov - Riesz Weil called, which also posts the mathematician Maurice René Fréchet and André Weil be appreciated also the name of Jacob Davidovich Tamarkin and AN Tulajkov be mentioned by some authors, the latter had treated the special case. Such compactness criteria have many applications, particularly in the theory of partial differential equations.

The sequence space lp

The situation for the sequence spaces is particularly easy to represent, the following theorem was proved for 1908 by Fréchet, :

A subset () is exactly then precompact if the following two conditions are met:

• . This is the -th component of.
• .

Function spaces Lp

Ornate are Kompaktheitskriteríen for Lp- spaces over non- discrete basic quantities. By means of reduction to the set of Arzelà - Ascoli, one can show:

Set of Kolmogorov - Riesz: A subset is precompact if and only if

• Is confined in the norm,
• .

It is outside of the unit interval to form in the above formula can. An analogous set of course applies for arbitrary.

An extension of this theorem to unbounded domains requires an additional condition:

Set of M. Riesz: A subset () is precompact if and only if

• Is confined in the norm,
• ,
• .

It stands for the ball around 0 with radius.

Locally Compact Abelian Groups

The theorem of M. Riesz can not be generalized to Lp- spaces over arbitrary measure spaces, since is made in the second condition of the compactness criterion of the addition and thus of the group structure of the utility. Let now a locally compact abelian group and was on a hair cal measure. If a Banach space, so you can as above with the space of all measurable functions. The standard makes it a Banach space. This obviously generalizes the above considered spaces. Instead of balls around 0, here we consider a directed network with respect to the union of compact sets in, so that every compact set in an amount of from is included.

Nicolae Dinculeanu has proved the following generalization of the above compactness criterion:

Set: A subset (, locally compact abelian group, Banach space ) is precompact if and only if

• For all measurable subsets is precompact
• ,
• .

This version has been shown in the case of scalar functions so, M. Riesz. One goes back to Kolmogorov and JD Tamarkin version that uses an approximation of the one was also generalized by N. Dinculeanu on the Banach space -valued case. For the following discussion of this result, a Nullumbegungsbasis of relatively compact open sets is in. For each choose a function that limited, measurable Us symmetric ( ie ) with support in and. One may choose, for example, wherein the characteristic is a function of. Pros and convolution is defined. Then, and; That is, the power is an approximation In this sense of unity. We have the following

Set: A subset (, locally compact abelian group, Banach space ) is precompact if and only if

• For all measurable subsets is precompact
• ,
• .

In the earlier versions and for the networks have been and used. Applying this rate to the locally compact abelian group, then the first condition is equivalent to, because a lot of finite measure is finite; the second condition is empty if you select the network, and the last condition becomes, if one sets. With a suitable isomorphism between and you get exactly the cited theorem on spaces.

Further generalizations

Further generalizations to non-commutative locally compact groups were found by Josh Isralowitz. An extension of compactness criteria for this type on other defined on locally compact groups function spaces can be found in Hans G. Feichtinger