Hello everyone. I'm in a little trouble trying to find the proof of a theorem stated by W. T. Gowers. It is the Lemma 1.6 in his article 'An infinite Ramsey theorem and some Banach space dichotomies' (Annals of Mathematics, 156 (2002)). He refers Lindenstrauss but I couldn't find somehting clear and detailed about the proof either.
If anyone knows about another reference about this proof I would appreciate it.


Thanks.



A proof about an unconditional basis theorem
Dan, I doubt that you will find a proof of this in the literature. To get (ii) from (i), note that (i) gives you a block basic sequence $x\_1,....,x\_m$ in the block subspace $Y$ s.t. $\|\sum\_{i=1}^m x\_i\|= 1$ and for some choice $a\_i$ of signs, 


$\|\sum\_{i=1}^m a\_i x\_i\| > C$. WLOG $a\_1=-1$. Now group together maximal blocks all of whose signs are 


the same to rewrite 
$\sum\_{i=1}^m a\_i x\_i$ as $\sum\_{j=1}^n (-1)^j y\_j$, where $y\_j$ is the sum of the $x\_i$'s in the $j$-th maximal block. 


I assumed real coefficients in the above, but the complex case is only a bit more involved.


If this explanation isn't sufficient, I suggest that you study further the section in [LT] on bases. I addressed the only point that I thought might be troubling to someone who had studied [LT].


