Problem in Banach space - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T21:47:45Z http://mathoverflow.net/feeds/question/30374 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/30374/problem-in-banach-space Problem in Banach space Dan 2010-07-03T05:30:19Z 2010-07-03T05:37:33Z <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 $&sigma;$ 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 $&sigma;$. 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>