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.



Stone-Weierstrass, uniform convergence, and sums
As I said in the comment, this is clearly not true in general: the algebra of polynomial function on $[0,1]$ is dense among all continuous functions and generated by the $x^i$ but only functions that are analytic on the unit disk can be sum of a series of the form $\sum\_i a\_i x^i$.


Now there is indeed a few case where it works:


If you have a scalar product on your space of functions, that the $F\_i$ are orthogonal for this scalar product and that the space they generates is also dense for the norm induced by the scalar product (so for example if uniform convergence implies convergence for the scalar product) then your function can be approximated by a series $\sum a\_i F\_i$ which converge normally in the sense of the scalar product (but not necessarily uniformly).


This is very easy to prove:any element will be the limit of its projection on the succesive space generated by the first $n$ generators and the orthogonality relation between the generator ensure that the successive projection are the partial sum of a series.


This apply for example to the case of the algebra of trigonometric functions which gives Fourier series. In this case of course there is some well known trick involving convolution by well chosen kernel to recover uniform convergence under some additional conditions.


There is of course other kind of example: what you are looking for in general is called a [Schauder basis](https://en.wikipedia.org/wiki/Schauder_basis).


