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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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This is the negation that $\{f_{n}\}_n$ is a Bessel sequence and the negation do not make sense if $f=0$ because both sides are zero. –  user29999 Jun 7 '12 at 14:04
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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! –  Kim Jun 7 '12 at 14:10

1 Answer 1

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.

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