Friday, 28 May 2010

banach spaces - 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 x1,....,xm in the block subspace Y s.t. |summi=1xi|=1 and for some choice ai of signs,



|summi=1aixi|>C. WLOG a1=1. Now group together maximal blocks all of whose signs are



the same to rewrite
summi=1aixi as sumnj=1(1)jyj, where yj is the sum of the xi'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].

No comments:

Post a Comment