Weakly null sequences in 𝐿₁

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 ε > 0 \varepsilon >0 , the Haar basis is ( 1 + ε ) (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.