Problem in Banach space - MathOverflow most recent 30 from http://mathoverflow.net2013-05-23T21:47:45Zhttp://mathoverflow.net/feeds/question/30374http://www.creativecommons.org/licenses/by-nc/2.5/rdfhttp://mathoverflow.net/questions/30374/problem-in-banach-spaceProblem in Banach spaceDan2010-07-03T05:30:19Z2010-07-03T05:37:33Z
<p>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.</p>
<p>It says:</p>
<blockquote>
<p>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 x<sub>i</sub> of norm 1. Suppose (y'<sub>1</sub>, z'<sub>1</sub>, y'<sub>2</sub>, z'<sub>2</sub>,...,y'<sub>n</sub>, z'<sub>n</sub>) is in $σ$. Prove that the norms
<code>$\Vert \sum_{i=1}^n (y'_i + z'_i)\Vert$</code> and <code>$\Vert \sum_{i=1}^n (y_i' - z'_i)\Vert $</code>
are both at least 1/2.</p>
</blockquote>
<p>Any help is welcome. </p>
<p>Thanks.</p>