Is every unit norm Bessel sequence in a Hilbert space a finite union of separated ones? Is every unit norm separated sequence a finite union of uniformly minimal (minimal with uniformly bounded biorthogonal vectors) ones?
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1$\begingroup$ It might be worth defining these terms... $\endgroup$– Matthew DawsCommented Sep 15, 2010 at 18:38
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$\begingroup$ Unit norm means each vector is of norm 1. Separated means that there is a constant $c>0$ s.t. the distance between any two vectors is $>c$. Minimal means that none of the vectors is in the closed span of the others. $\endgroup$– MiMCommented Sep 15, 2010 at 19:23
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$\begingroup$ And a Bessel sequence? $\endgroup$– Yemon ChoiCommented Sep 15, 2010 at 21:57
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$\begingroup$ $f_n$ is a Bessel sequence if $\sum|<f|f_n>|^2\leq \|f\|^2$ for all vectors $f$. $\endgroup$– MiMCommented Sep 15, 2010 at 22:21
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$\begingroup$ instead of $\leq \|f\|^2$ it should be $\leq C\|f\|^2$. Sorry. $\endgroup$– MiMCommented Sep 15, 2010 at 22:23
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1 Answer
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Your questions are weakenings of the Feichtinger conjecture, which is equivalent to the Kadison-Singer problem. See
http://www.aimath.org/WWN/kadisonsinger/FrameProblems.pdf
and the references therein.
Your second question is Problem 2.2 there.
The questions themselves are not obvious ones. Why did you ask them?
answered Sep 16, 2010 at 3:32