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Hi everybody, I've got an exercise about Banach spaces and I can't see how to solve it. It is a very simple problem and I know it might be some little detail I'm missing, and that is why I'm asking for help.

It says:

Let X be a Banach space with a monotone basis. Let $σ$ be the set of all finite block bases in the unit ball of X that contain at least one vector xi of norm 1. Suppose (y'1, z'1, y'2, z'2,...,y'n, z'n) is in $σ$. Prove that the norms $\Vert \sum_{i=1}^n (y'_i + z'_i)\Vert$ and $\Vert \sum_{i=1}^n (y_i' - z'_i)\Vert $ are both at least 1/2.

Any help is welcome.


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Dan, Is the exercise for credit? If not let us know, I can give you a useful hint. –  Gideon Schechtman Jul 3 '10 at 9:06
No it isn't. It is part of a series of lemmas I'm trying to stablish for proving a theorem in other way. I would appreciate the hint, thanks for your interest. –  Dan Jul 3 '10 at 15:12
Hint: Prove first that the norm of the natural projection onto any tail of the basis is of norm at most 2. –  Gideon Schechtman Jul 3 '10 at 19:18
When you say 'tail', you mean...? –  Dan Jul 4 '10 at 2:02
Let $y_1,y_2,\dots$ be the monotone basis. Let $P_k$ be the projection onto the span of $y_1,y_2,\dots,y_k$ which annihilates $y_{k+1},y_{k+2},\dots$. Monotonicity means that $\|P_k\|\le 1$. It follows that $\|I-P_k\|\le 2$. The hint is to use this fact. – Gideon Schechtman 14 secs ago –  Gideon Schechtman Jul 4 '10 at 6:41
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