Geodesic
Weakly null sequences in 𝐿₁
Johnson, William1 ; Maurey, Bernard2 ; Schechtman, Gideon3
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

Voir la notice de l'article provenant de la source American Mathematical Society

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 
@article{10_1090_S0894_0347_06_00548_0,
     author = {Johnson, William and Maurey, Bernard and Schechtman, Gideon},
     title = {Weakly null sequences in {\dh}{\textquestiondown}\^a‚},
     journal = {Journal of the American Mathematical Society},
     pages = {25--36},
     publisher = {mathdoc},
     volume = {20},
     number = {1},
     year = {2007},
     doi = {10.1090/S0894-0347-06-00548-0},
     url = {http://geodesic.mathdoc.fr/articles/10.1090/S0894-0347-06-00548-0/}
}
TY  - JOUR
AU  - Johnson, William
AU  - Maurey, Bernard
AU  - Schechtman, Gideon
TI  - Weakly null sequences in 𝐿₁
JO  - Journal of the American Mathematical Society
PY  - 2007
SP  - 25
EP  - 36
VL  - 20
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/articles/10.1090/S0894-0347-06-00548-0/
DO  - 10.1090/S0894-0347-06-00548-0
ID  - 10_1090_S0894_0347_06_00548_0
ER  - 
%0 Journal Article
%A Johnson, William
%A Maurey, Bernard
%A Schechtman, Gideon
%T Weakly null sequences in 𝐿₁
%J Journal of the American Mathematical Society
%D 2007
%P 25-36
%V 20
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/articles/10.1090/S0894-0347-06-00548-0/
%R 10.1090/S0894-0347-06-00548-0
%F 10_1090_S0894_0347_06_00548_0
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

[1] Gowers, W. T., Maurey, B. The unconditional basic sequence problem J. Amer. Math. Soc. 1993 851 874

[2] Kadec, M. I., Peå‚Czyå„Ski, A. Bases, lacunary sequences and complemented subspaces in the spaces 𝐿_{𝑝} Studia Math. 1961/62 161 176

[3] Ledoux, Michel, Talagrand, Michel Probability in Banach spaces 1991

[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] Peå‚Czyå„Ski, 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

Cité par Sources :