Difference between frames and complete sequence Let $\mathcal{H}$ be a separable Hilbert space with inner product $\langle\cdot, \cdot\rangle$.
A sequence $\left\{f_k\right\}_{k=1}^{\infty}$ of elements in $\mathcal{H}$ is a frame for $\mathcal{H}$ if there exist constants $A, B>0$ such that
$$
A\|f\|^2 \leq \sum_{k=1}^{\infty}\left|\left\langle f, f_k\right\rangle\right|^2 \leq B\|f\|^2, \quad \forall f \in \mathcal{H} .
$$
It follows from this definition that if $\left\{f_k\right\}_{k=1}^{\infty}$ is a frame for $\mathcal{H}$, then
$$
\overline{\operatorname{span}}\left\{f_k\right\}_{k=1}^{\infty}=\mathcal{H} .
$$
i.e. the sequence $\left\{f_k\right\}_{k=1}^{\infty}$ is complete. Is the converse still true? What is a sequence that is complete but it is not a frame?

Consider $f_n = \frac{1}{n}e_n$, where $e_n \in \ell^2(\mathbb{N})$ is the $n$-th standard basis vector. It is easy to see that this is complete but not a frame (do it!).
