This is likely to be asked in the past, but I cannot find any answers about how to prove $B=\{e^{ikx}:k\in \mathbb Z\}$ is an orthogonal basis of $L^2[-\pi , \pi]$. I find a proof here. This proof is very computational.
Are there any other less computational proofs that doesn't rely on much algebraic calculations? For example, can I prove that the span of $B$ is dense by using the result that if $\langle x,e\rangle=0,\forall e\in B$ then $B$ is an orthonormal basis, or something like that?