I'm trying to prove the following:
Let $f=\sum_{n=-N}^{n=N}c_n e_n=c_n e^{2\pi i nx} (n\in\mathbb{Z},x\in\mathbb{R})$ be a trigonometric polynomial. Then we have $c_n:=\langle f,e_n\rangle$ for all $-N\leq n\leq N$ and $\langle f,e_n\rangle =0$ if $n > N$ or $n < -N.$
Also, we have the identity $\|f\|_2 ^2 =\sum_{n=-N}^{N} |c_n|^2$."
Now, I've managed to prove the first two identities and also that $\langle e_n,e_m\rangle =1$ if $n=m$, $\langle e_n,e_m\rangle =0$ if $n\neq m$ and $\|e_n\|=1$ but I haven't yet managed to prove that $\|f\|_2 ^2 =\sum_{n=-N}^{N} |c_n|^2$ (I tried by substituting $\ f=\sum_{n=-N}^{n=N}c_n e_n$ in the definition $\|f\|_2:=\sqrt{\langle f,f\rangle }=(\int_{[0,1]}f(x)\overline{f(x)}dx)^{1/2}=(\int_{[0,1]}|f(x)|^2 dx)^{1/2}$ but without success) so I'd appreciate any hint.