Does a Bessel sequence not being a frame mean that $\sum_n|\langle f,f_n\rangle|^2>B\|f\|^2$ for some $f\in H$?

Question

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)

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.