Rewriting elements of the closure of a countable set as an infinite sum of the spanning element in Hilbert spaces.

Question

Let $H$ be a Hilbert space and $f \in \overline{span\{\phi_{m} : m \in \mathbb{Z}\}}$ with $\phi_m \in H$ for every $m \in \mathbb{Z}$ . Why can we write \begin{equation} f = \sum_{m \in \mathbb{Z}} c_m \phi_m \end{equation} for some coefficients $c_m \in \mathbb{R}$?

Answer 1

Let $\lbrace h_n \rbrace$ be a sequence in the span converging to $f$. Then $$h_n = \sum_{i} c_{i,n} \phi_{i}$$ for coefficients $c_{i,n}$ and elements $\phi_{i,n}$ (for each $n$, only finitely many $c_{i,n}$ are nonzero). It is checked that the convergence of $h_n$ implies that $c_{i,n}$ converges (in $n$) for each $i$. Thus if $$c_i = \lim_{n \rightarrow \infty} c_{i,n},$$ then $f = \sum c_i \phi_i$.