Assume $\{F_n\}_{n\in \mathbb{N}} \subset C(X)$ is some series of continuous complex functions on the compact Hausdorff space $X$. Assume also that the $F_n$ separate the points of $X$, and that the $F_n$ have the property that every product can be written as a finite sum, i.e. $F_{n_1} F_{n_2} = \sum_{k=1}^K a_k F_k$.

Let $\mathcal{A} = \mathsf{span} \{F_n | n\in\mathbb{N}\}$ be the complex linear span of $\{F_n\}_n$. Because products can be written as sums, the span, $\mathcal{A}$, is the subalgebra of $C(X)$ generated by the $F_n$.

By Stone-Weierstrass, $\mathcal{A}$ is a dense subset of $C(X)$ under the sup-norm, i.e. any function $f\in C(X)$ can be written as the uniformly convergent limit of a sequence $\{f_j\}_j$ in $\mathcal{A}$. This is all well and good, but what I would really like is to write each $f\in C(X)$ as an infinite linear combination of the $F_n$.

That is, each of the $f_j$ in the sequence that uniformly converges to $f$ is a finite linear combination $f_j = \sum_{k=1}^{K_n} a_{j,k}\, F_k$, but does that imply that I can write the limit, $f$, as an infinite sum $f = \sum_{k=1}^{\infty} a_k F_k$ - and how? If not, are there some extra conditions that I can impose to make this possible?

Many thanks in advance.