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I'm posting this question again because I'm still confused about the answer!

A sequence $\{f_{n}\}_{n\in I}$ is called a Bessel sequence in a Hilbert space $H$, if there exists $B>0$ such that $\sum_{n\in I}|\langle f,f_{n}\rangle|^{2}\leq B\|f\|^{2}$ for all $f\in H$.

Now my questin is: if a given sequence is not Bessel sequence, does it mean that

given $B>0$, there exists (a non-zero) $f\in H$ such that $ \sum_{n\in I}|\langle f,f_{n}\rangle|^{2}> B\|f\|^{2}$

Thanks!

(old post: If a sequence is not a frame)

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    So just to be sure: if we pick another $B$ we would find another $f\in H$ satisfying the second relation, and so forth, right!2012-06-07

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The sequence $\{f_n\}$ is not a Bessel sequence if for all integer $N$, we can find $g_N\in H$ such that for all integer $N$, $\sum_{n\in I}|\langle g_N,f_n\rangle|^2> N \lVert g_N\rVert^2,$ exactly what you mean.