## Weakly null sequences in $L_1$

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- by William B. Johnson, Bernard Maurey and Gideon Schechtman
- J. Amer. Math. Soc.
**20**(2007), 25-36 - DOI: https://doi.org/10.1090/S0894-0347-06-00548-0
- Published electronically: September 19, 2006

## Abstract:

We construct a weakly null normalized sequence $\{f_i\}_{i=1}^{\infty }$ in $L_1$ so that for each $\varepsilon >0$, the Haar basis is $(1+\varepsilon )$-equivalent to a block basis of every subsequence of $\{f_i\}_{i=1}^{\infty }$. In particular, the sequence $\{f_i\}_{i=1}^{\infty }$ has no unconditionally basic subsequence. This answers a question raised by Bernard Maurey and H. P. Rosenthal in 1977. A similar example is given in an appropriate class of rearrangement invariant function spaces.## References

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## Bibliographic Information

**William B. Johnson**- Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843-3368
- MR Author ID: 95220
- Email: johnson@math.tamu.edu
**Bernard Maurey**- Affiliation: Laboratoire d’Analyse et de Mathématiques Appliquées, UMR CNRS 8050, Université de Marne la Vallée, 77454 Champs-sur-Marne, France
- Email: maurey@univ-mlv.fr
**Gideon Schechtman**- Affiliation: Department of Mathematics, Weizmann Institute of Science, Rehovot, Israel
- MR Author ID: 155695
- Email: gideon.schechtman@weizmann.ac.il
- Received by editor(s): June 8, 2004
- Published electronically: September 19, 2006
- Additional Notes: The first author was supported in part by NSF grant DMS-0200690 and NSF grant DMS-0503688, Texas Advanced Research Program 010366-0033-20013 and the U.S.-Israel Binational Science Foundation.

The last author was supported in part by the Israel Science Foundation and the U.S.-Israel Binational Science Foundation and was a participant in the NSF Workshop in Linear Analysis and Probability, Texas A&M University. - © Copyright 2006 by the authors
- Journal: J. Amer. Math. Soc.
**20**(2007), 25-36 - MSC (2000): Primary 46B15, 46E30
- DOI: https://doi.org/10.1090/S0894-0347-06-00548-0
- MathSciNet review: 2257395