Re: Compactness Criteria for L^p Spaces

From: Zdislav V. Kovarik (kovarik_at_mcmaster.ca)
Date: 12/12/04 

Date: Sat, 11 Dec 2004 21:31:42 -0500

On Sun, 12 Dec 2004, Kira Yamato wrote:

> Does anyone know (or can give reference) to a necessary and sufficient
> condition for compactness in L^p spaces? In particular, I need to know
> when it is true that
>
> Given a subspace E in L^p for a fixed p (1<=p<inf), any sequence in E
> has a converging subsequence in L^p.
>
> I know there is the closely-related Ascoli theorem, but that one is
> about the subspace of continuous functions in the sup-norm function
> space. The condition there is that the subspace is closed and bounded
> with equi-continuity.
>
> Is there a similar condition for subspace of L^p spaces?
>
> Thanks.
> Kira.

 Dunford-Schwartz: Linear Operators I, Theorems IV.8.18 and IV.8.20 (the
latter one is for the real axis, and looks like a version of Arzela-Ascoli
Theorem). It is too messy to render in plain text.
 Weak sequential compactness: in reflexive spaces, it is equivalent to
boundedness (with weak closedness)(same book, II.3 28).

Cheers, ZVK(Slavek).

Quantcast