Q1) In a course, it's written that $L^2(0,1)$ is spanned by $\{e^{2\pi inx}\}_{n\in\mathbb Z}$. How can I show it ?
Q2) Let $f\in L^2(0,1)$. Then we have Parseval equality, i.e. $$\left\|f\right\|^2=\sum_{k=-\infty }^\infty \left|c_i\right|^2,$$ where $c_i$ are the Fourier coefficient. Is this convergence in the $L^2-$sense ? I mean, is $$\|f\|^2=\lim_{N\to \infty }\sum_{k=-N}^N\left|c_i\right|^2$$ in the $L^2-$sense ?