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)