# 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