A proof about an unconditional basis theorem

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.

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For those of us who don't have access to the article, would you mind stating the Lemma? – Qiaochu Yuan Mar 17 '10 at 18:44
arxiv.org/abs/math.FA/0501105 – Mark Meckes Mar 17 '10 at 19:28
This article also refers to arxiv.org/abs/math/9205204, which, like Lindenstrauss and Tzafriri, does not state the lemma explicitly in that form. – Mark Meckes Mar 17 '10 at 19:45

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].

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I'm not a Banach-space specialist, and don't have a copy of Lindenstrauss-Tzafriri to hand (though maybe the Albiac-Kalton book would be a friendlier read, if your library has a copy?) but it seems to me quite natural to take the contrapositive and try to work with that. In other words, show that TFAE

1) X has a (closed) subspace with an unconditional basis.

2) There exists a block subspace Y of X and a real number C such that for every sequence $y_1 < y_2 < \dots < y_n$ in $Y$ we have the inequality

$$\left\Vert \sum_{i=1}^n y_i \right\Vert \leq C \left\Vert \sum_{i=1}^n (-1)^i y_i \right\Vert$$

Certainly if we assume 2) then there is a natural candidate for the subspace desired in 1) ...

Maybe it would help if you could show us a more precise or more localised part of the problem which you're stuck on -- is it to do with the equivalent definitions of unconditional basis, or something else?

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