I am self-studying A Course in Functional Analysis by John Conway. In Chapter 1, I saw the following example:
Let $\mathcal{H}=L^2_{\mathbb{C}} [0,2\pi]$ and for n in $\mathbb{Z}$ define $e_n$ in $\mathcal{H}$ by $e_n(t)=(2\pi)^{-1/2} e^{int}$. Then $\{e_n:n \in \mathbb{Z}\}$ is an orthonormal set in $\mathcal{H}$. Here $L^2_{\mathbb{C}} [0,2\pi]$ is the space of complex-valued square integrable functions.
Related definition: An orthonormal subset of a Hilbert space $\mathcal{H}$ is a subset $\mathcal{E}$ having the properties: (a) for e in $\mathcal{E}$, $\Vert e \Vert =1$; (b) if $e_1,e_2 \in \mathcal{E}$ and $e_1 \neq e_2$, then $e_1 \perp e_2$.
I am trying to verify this example,specificaly $\{e_n:n \in \mathbb{Z}\}$ is an orthonormal set in $\mathcal{H}$, but I am not sure how to construct. Any thought provoking idea is appreciated.
Note that you aren't trying to show that this is a Hilbert Basis (IE an orthonormal set with the additional feature that every vector can be written as a countable linear combination of elements within). So all you have to do is show that A: Each one is normal, ie $$\int_0^1e_n\overline{e_n}dt=1$$ and that they are pairwise orthogonal, i.e for $n\neq m$, $$\int_0^1e_n\overline{e_m}dt=0$$