Give an example of a spanning set of $\ell^2(N)$ which is also a Bessel sequence but not a frame for $\ell^2(N)$

Question

We know that in a finite dimensional Hilbert space, every spanning set is a frame, but this is not true for infinite dimensional space. It is easy to find an example which is a spanning set but not a Bessel sequence. But i am trying to find an example of a sequence in $\ell^2(N)$ which is a Bessel sequence, but not a frame.

Answer 1

Yes, the sequence $\{f_n=e_n/n\}$ is a Bessel sequence, but is not a frame, since for $v=e_k$ we have $\sum_n|\langle v,f_n\rangle|^2 = 1/k^2$, which is not bounded below by $A\|v\|^2$ for any fixed $A>0$.

The above set is spanning. As far as I know, a Bessel sequence is not required to span the space. In this case, $\{0,0,0,\dots\}$ is a trivial example of a Bessel sequence that is not a frame.

Related questions, just for completeness:
Bessel sequence in Hilbert space
If a sequence is not a frame