I'm working on a problem dealing with complete sequences on Hilbert spaces. An orthonormal sequence $(e_n)$ in a Hilbert space $H$ is called complete if the only member of $H$ orthogonal to each $e_n$ is the zero vector.
I know that an orthonormal sequence $(e_n)$ is complete if and only if $\|x\|^2=\sum_1^\infty|\langle x,e_n\rangle|^2$ for all $x\in H$. Further, for any $x\in H$ we can write $$ x=\sum_{n=1}^\infty\langle x,e_n\rangle e_n. $$ Below is the problem I am tackling:
Prove that $(e^{2\pi inx})^\infty_{n=-\infty}$ is a complete orthonormal sequence in $L^2(0,1)$.
I have shown that $(e^{2\pi inx})$ is orthonormal; that is, $\langle e^{2\pi inx},e^{2\pi imx}\rangle=\delta_{nm}$.
Now I am trying to show that it is complete by showing that if $x\in H$ and $\langle x,e^{2\pi inx}\rangle=0$ for every $n$, then $x=0$. Any suggestions on what to do next?
