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The Kadec-Pełczyński theorem in $ L^p$, $ 1\le p<2$

Authors: I. Berkes and R. Tichy
Journal: Proc. Amer. Math. Soc. 144 (2016), 2053-2066
MSC (2010): Primary 46B09, 46B25
Published electronically: September 15, 2015
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Abstract: By a classical result of Kadec and Pełczyński (1962), every normalized weakly null sequence in $ L^p$, $ p>2$, contains a subsequence equivalent to the unit vector basis of $ \ell ^2$ or to the unit vector basis of $ \ell ^p$. In this paper we investigate the case $ 1\le p<2$ and show that a necessary and sufficient condition for the first alternative in the Kadec-Pełczyński theorem is that the limit random measure $ \mu $ of the sequence satisfies $ \int _{\mathbb{R}} x^2 d\mu (x)\in L^{p/2}$.

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Additional Information

I. Berkes
Affiliation: Institute of Statistics, Graz University of Technology, Kopernikusgasse 24, 8010 Graz, Austria

R. Tichy
Affiliation: Institute of Mathematics A, Graz University of Technology, Steyrergasse 30, 8010 Graz, Austria

Received by editor(s): September 8, 2014
Received by editor(s) in revised form: May 27, 2015
Published electronically: September 15, 2015
Additional Notes: The research of the first author was supported by FWF grant P24302-N18 and OTKA grant K 108615.
The research of the second author was supported by FWF grant SFB F5510.
Communicated by: Thomas Schlumprecht
Article copyright: © Copyright 2015 American Mathematical Society