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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

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The Kadec-Pełczyński theorem in $L^p$, $1\le p<2$
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by I. Berkes and R. Tichy PDF
Proc. Amer. Math. Soc. 144 (2016), 2053-2066 Request permission


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
  • MR Author ID: 35400
  • Email:
  • R. Tichy
  • Affiliation: Institute of Mathematics A, Graz University of Technology, Steyrergasse 30, 8010 Graz, Austria
  • MR Author ID: 172525
  • Email:
  • 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
  • © Copyright 2015 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 144 (2016), 2053-2066
  • MSC (2010): Primary 46B09, 46B25
  • DOI:
  • MathSciNet review: 3460166