Weakly null sequences in L 1 L_1 Academic Article

We construct a weakly null normalized sequence { f i } i = 1 {f_i}_{i=1}^{infty } in L 1 L_1 so that for each > 0 varepsilon >0 , the Haar basis is ( 1 + ) (1+varepsilon ) -equivalent to a block basis of every subsequence of { f i } i = 1 {f_i}_{i=1}^{infty } . In particular, the sequence { f i } i = 1 {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.