Weakly null sequences in 𝐿₁

Johnson, William; Maurey, Bernard; Schechtman, Gideon

doi:10.1090/S0894-0347-06-00548-0

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

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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.

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