# Bessel sequence, uniformly minimal, separated

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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It might be worth defining these terms... – Matthew Daws Sep 15 '10 at 18:38
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. – MiM Sep 15 '10 at 19:23
And a Bessel sequence? – Yemon Choi Sep 15 '10 at 21:57
$f_n$ is a Bessel sequence if $\sum|<f|f_n>|^2\leq \|f\|^2$ for all vectors $f$. – MiM Sep 15 '10 at 22:21
instead of $\leq \|f\|^2$ it should be $\leq C\|f\|^2$. Sorry. – MiM Sep 15 '10 at 22:23

Your questions are weakenings of the Feichtinger conjecture, which is equivalent to the Kadison-Singer problem. See

and the references therein.

Your second question is Problem 2.2 there.

The questions themselves are not obvious ones. Why did you ask them?

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