I want to know, why $\{e_n\mid n\in\mathbb{Z}\}$ is an orthonormal basis of $L^2(\mathbb{T})$, where $\mathbb{T}=\{z\in\mathbb{C}\mid |z|=1\}$, $e_n(z)=z^n$, and $\int_{\mathbb{T}} f(z)\,dz:=\int_0^1f(e^{2\pi i t})\,dt$. I asked here Orthonormal basis of $L^2(T)$ why this set is orthonormal, and I wanted to ask, how to prove $\overline{\operatorname{span}\{e_n\mid n\in\mathbb{Z}\}}=L^2(\mathbb{T}),$ but I forgot to set an password and now I can't log in again and edit my question. (I promise, I don't do this again)
Therefore I open a new question, sorry. Why is $\overline{\operatorname{span}\{e_n\mid n\in\mathbb{Z}\}}=L^2(\mathbb{T})$?
$\underline{Hint}$: To show that $\{ e_n\mid n\in\mathbb{Z}\}$ is an orthonormal basis (ONB) for $L^2(\mathbb{T})$, we must show that $\operatorname{span}\{e_n\mid n\in\mathbb{Z}\}$ is dense in $^2(\mathbb{T})$, which is equivalent to stating that $\exists g\in \operatorname{span}\{ e_n\mid n\in\mathbb{Z}\}$ such that $$ \|f-g\|_{L^2(\mathbb{T})}=\left( \int^{\pi}_{-\pi}|f(x)-g(x)|^2dx \right)^{\frac{1}{2}}<\epsilon . $$ To prove this, show that every $f\in C(\mathbb{T})$ can be approximated arbitrarily well by functions from $\operatorname{span}\{e_n\mid n\in\mathbb{Z}\}$ with respect to the supremum norm. The supremum norm is stronger than the $L^2$ norm which implies that we can approximate $f\in C(\mathbb{T})$ arbitrarily well by functions from $\operatorname{span}\{e_n\mid n\in\mathbb{Z}\}$ with respect to the $L^2$ norm.
Observing that $C(\mathbb{T})$ is dense in $L^2(\mathbb{T})$ (with $L^2$ norm), this implies that $\operatorname{span}\{ e_n\mid n\in\mathbb{Z} \}$ is dense in $L^2(\mathbb{T})$ (with $L^2$ norm). Therefore, every function in $L^2(\mathbb{T})$ can be approximated by a Fourier series with respect to the $L^2$ orthonormal set of complex trigonometric polynomials $\{e_n\mid n\in\mathbb{Z}\}$. The result follows.
Alternatively, if you can prove that $$ \overline{\operatorname{span} \{e_n\mid n\in\mathbb{Z}\}}\subset L^2(\mathbb{T})~\text{and}~L^2(\mathbb{T})\subset\overline{\operatorname{span} \{e_n\mid n\in\mathbb{Z}\}} $$ hold, then the result will follow.