I'm a little confuse about this problem:
I have two sequences $\{a_{n}\}$ and $\{b_{n}\}$ in a Hilbert space $H$, with $\{a_{n}\}$ is a Bessel sequence for $H$ if and only if $\{b_{n}\}$ is a Bessel sequence for $H$.
The problem is that: If $\{a_{n}\}$ has Bessel bound $\geq K$, for every $K\in \mathbb N$, does this mean that $\{a_{n}\}$ is not Bessel sequence, and hence $\{b_{n}\}$ is also not Bessel sequence?
Definition: A sequence $\{a_{n}\}$ is a Bessel sequence in $H$ with Bessel bound $B>0$ if $$ \sum_{n} |\langle f,a_{n}\rangle|^{2}\leq B ||f||^{2} $$ for all $f\in H$.