Geodesic
Weakly null sequences in 𝐿₁
Johnson, William 1   ; Maurey, Bernard 2   ; Schechtman, Gideon 3
1 Department of Mathematics, Texas A&M University, College Station, Texas 77843-3368
2 Laboratoire d’Analyse et de Mathématiques Appliquées, UMR CNRS 8050, Université de Marne la Vallée, 77454 Champs-sur-Marne, France
3 Department of Mathematics, Weizmann Institute of Science, Rehovot, Israel
Journal of the American Mathematical Society, Tome 20 (2007) no. 1, pp. 25-36 Cet article a éte moissonné depuis la source American Mathematical Society

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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.
DOI : 10.1090/S0894-0347-06-00548-0

Johnson, William  1   ; Maurey, Bernard  2   ; Schechtman, Gideon  3

1 Department of Mathematics, Texas A&M University, College Station, Texas 77843-3368
2 Laboratoire d’Analyse et de Mathématiques Appliquées, UMR CNRS 8050, Université de Marne la Vallée, 77454 Champs-sur-Marne, France
3 Department of Mathematics, Weizmann Institute of Science, Rehovot, Israel 
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Johnson, William; Maurey, Bernard; Schechtman, Gideon. Weakly null sequences in 𝐿₁. Journal of the American Mathematical Society, Tome 20 (2007) no. 1, pp. 25-36. doi: 10.1090/S0894-0347-06-00548-0

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[4] Lindenstrauss, Joram, Tzafriri, Lior Classical Banach spaces. II 1979

[5] Maurey, B., Rosenthal, H. P. Normalized weakly null sequence with no unconditional subsequence Studia Math. 1977 77 98

[6] Maurey, B., Schechtman, G. Some remarks on symmetric basic sequences in 𝐿₁ Compositio Math. 1979 67 76

[7] Pełczyński, A., Semadeni, Z. Spaces of continuous functions. III. Spaces 𝐶(Ω) for Ω without perfect subsets Studia Math. 1959 211 222

[8] Rosenthal, Haskell P. On subspaces of 𝐿^{𝑝} Ann. of Math. (2) 1973 344 373

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