I have read some papers that make use of a 'Fourier basis' when decomposing some functions, but I am wondering whether there is more depth behind that phrase and what can we rigorously say about this basis. Suppose we want to be rigorous. In that case is it ok to simply say that the set of functions
$S = \{e^{i n \theta}: n \in \mathbb{Z}\},$
forms a basis for $L^2([0, 2\pi])$ because any function $f \in L^2([0, 2\pi])$ can be represented as a linear combination of cosines and sines? Or is it necessary to show that the set of functions $S$ is a Hamel or Schauder basis for the space $L^2([0, 2\pi])$? If it's the latter, which type of basis is this set of functions $S$, Hamel or Schauder? And how do we show it is a basis?
The fact that your set forms an orthonormal system can be verified directly by a calculation. The hard thing is to show that this system is complete (or, in other words, that $\operatorname{span}(S)$ which is the space of trigonometric polynomials is dense in $L^2([0,2\pi])$ with respect to the $L^2$ norm). One way to do it is to use first the Stone-Weierstrass theorem to show that the space of trigonometric polynomials is dense in $C[0,2\pi]$ with respect to the supremum norm and then use the density of continuous functions in $L^2([0,2\pi])$ (with respect to the $L^2$ norm) and the fact that converges in the supremum norm implies converges in the $L^2$ norm to deduce the required result.