Fréchet–Kolmogorov theorem - Alchetron, the free social encyclopedia

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 L^p space. It can be thought of as an L^p version of the Arzelà–Ascoli theorem, from which it can be deduced. The theorem is named after Maurice René Fréchet and Andrey Kolmogorov.

Statement

Let B be a bounded set in L p ( R n ) , with p ∈ [ 1 , ∞ ) .

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

lim r → ∞ ∫ | x | > r | f | p = 0 uniformly on B,
lim a → 0 ∥ τ a f − f ∥ L p ( R n ) = 0 uniformly on B,

where τ a f denotes the translation of f by a , that is, τ a f ( x ) = f ( x − a ) .

The second property can be stated as ∀ ε > 0 ∃ δ > 0 such that ∥ τ a f − f ∥ L p ( R n ) < ε ∀ f ∈ B , ∀ a with | a | < δ .

References

Fréchet–Kolmogorov theorem Wikipedia

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