Say we have an Inner Product space $(X, \langle\cdot ,\cdot \rangle)$. Let $(e_k)$ be an orthonormal sequence, where $k \in I$. Then we know by Bessel's inequality that $$ \sum_1^\infty |\langle x,e_k\rangle|^2 \le \|x\|^2$$ The $\langle x,e_k\rangle$ are called Fourier coefficients of x (with respect to $e_k$). How do we then show that any $x \in X$ can have at most countably many non-zero Fourier coefficients?
First we suppose that the orthonormal family $(e_k)$ is uncountable, otherwise it's trivial. So my first question is, how can there exist uncountably many $e_k$? Is there an example?
What I thought could help is the following: form the coefficients $\langle x,e_k\rangle$. We know that for every $n=1,2,3,\ldots$ $$ \sum_1^n |\langle x,e_k\rangle|^2 \le \|x\|^2$$
But I couldn't make any progress.