Prove that a set is orthonormal on $L_2$

Question

I would like to prove that the set of elements:

\begin{equation} A_n(t)=\left\{\frac{1}{\sqrt{2\pi}}e^{int}\right\}_{n=-\infty}^{\infty} \end{equation}

is an infinite orthonormal set, on space $L_2[-\pi, \pi]$.

Which inner product should I use and which norm to prove the orthonormality?

Answer 1

The inner product is

$$<\phi_n,\phi^*_m>=\int_{-\pi}^{\pi} \left(\sqrt{\frac{1}{2\pi}}e^{int}\right)\,\,\left(\sqrt{\frac{1}{2\pi}}e^{-imt}\right)\,dt$$

which equals $0$ for $m \ne m$ and $1$ for $m=n$.