Suppose I have a (infinite-dimensional) Hilbert Space $H$ with a countable orthornormal basis $\{e_n\}_{n\in\mathbb{Z}_{\geq 0}}$. By definition, every element $x\in H$ can be expressed as an infinite sum $$ x=\sum_{n\geq 0} c_n e_n $$ Suppose now I know that I have some other set of vectors $\mathcal{B}=\{f_n\}_{n\in\mathbb{Z}_{\geq 0}}$ and that each $e_n$ can be expressed as a $finite$ linear combination of elements of $\mathcal{B}$, that is $$ e_n=\sum_{m\geq 0} a_{nm} f_m,\quad \forall n\geq 0 $$ with only finitely many $a_{nm}$ non-zero for a given $n$ and similarly I can express all $f_m$ as a finite linear combination of $e_n$: $$ f_n=\sum_{m\geq 0} b_{nm} e_m,\quad \forall n\geq 0 $$ with only finitely many $b_{nm}$ non-zero for a given $n$.
Question: Can every element $x\in H$ be written as $$ x=\sum_{n\geq 0} k_n f_n\quad ? $$ where each $|k_n|<\infty$ are some coefficients? I am simply asking about existence.
The progress I have made so far is that I can write $x$ as the following: $$ x=\sum_{m}\sum_{n} c_m a_{mn_m}f_{n_m} $$ To get a positive answer to the question I need to swap the sums, which is possible when the series converges absolutely. For the moment I cannot say anything about absolute convergence, but the series does converge to $x$. When else do I need for absolute convergence?
Update: In order to clarify certain details regarding the origin of this problem and some further constraints let me state the following.
Suppose I have an infinite-dimensional inner product space $V$, with (Hammel) bases $\mathcal{B_1}=\{e_n\}_{n\geq 0}$ and $\mathcal{B_2}=\{f_n\}_{n\geq 0}$. The $e_n$ are orthonormal with respect to the inner product but the $f_n$ are not (or rather need not be). I then construct the Hilbert spaces $H_1$ of all convergent linear combinations $$ \sum_{n\geq 0} a_n e_n,\quad |a_n|<\infty $$ and the Hilbert space $H_2$ of all convergent linear combinations $$ \sum_{n\geq 0} b_n f_n,\quad |b_n|<\infty $$ My question then becomes: Is $H_1=H_2$?
I think the answer is no. Consider the space $(c_{00}, \|\cdot\|_2)$ of all finitely-supported sequences equipped with the $\ell^2$-norm.
Let $(e_n)_n$ be the canonical vectors in $c_{00}$ and let $f_n = e_n + e_1$. Then $(e_n)_n$ is an orthonormal Hamel basis for $c_{00}$ and $(f_n)_n$ is a Hamel basis for $c_{00}$.
Now, the completion of $(c_{00}, \|\cdot\|_2)$ is the Hilbert space $\ell^2$.
The sum $\sum_{n=1}^\infty \frac1ne_n$ is an element of $\ell^2$ but it cannot be written as $\sum_{n=1}^\infty \alpha_nf_n$. Indeed, we get
$$\sum_{n=1}^\infty \frac1ne_n = \sum_{n=1}^\infty \alpha_nf_n = \left(\sum_{n=1}^\infty \alpha_n\right)e_1 + \sum_{n=2}^\infty \alpha_ne_n$$
It follows $\alpha_n = \frac1n$ for $n \ge 2$ but then $\sum_{n=1}^\infty \alpha_n$ does not converge.