I need to prove that $\{e_n\ |\ n \in \Bbb Z \}$ is an orthonormal basis for $L^2 (\Bbb S^1).$

Question

Let $\mathcal H = L^2 (\Bbb S^1).$ Define $e_n(z) = z^n,\ n \in \Bbb Z.$ Show that $\{e_n\ |\ n \in \Bbb Z \}$ is an orthonormal basis for $L^2 (\Bbb S^1) \simeq L^2 [0,1].$

I need to first show that $\displaystyle \int_{0}^{1} e^{2 \pi (m-n)it}\ dt = 0,$ for $m \neq n,$ which is very easy to show. Secondly I need to show that $\{e_n\ |\ n\in \Bbb Z \}$ is total. Can somebody please shed some light on how to show totality of the orthonormal system? Is it due to Fourier series expansion? Can every square summable function on $[0,1]$ admits Fourier series expansion? Any help or suggestion in this regard will be highly appreciated.

Thanks for your time.

Answer 1

Hint: Note that $C(S^1)$ is dense in $L^2(S^1)$. Then, use this fact together with the Stone-Weierstrass theorem to conclude.