Proposition 33.1.3.label Let $G$ be a compact Abelian group with $|G| = 1$, then $\wh G$ is an orthonormal set in $L^{2}(G; \complex)$.
Proof, [Proposition 4.4, Fol16]. Let $\phi, \psi \in \wh G$ be distinct, then there exists $x_{0} \in G$ with $\dpn{x_0, \phi}{G}\dpn{x_0, \psi}{G}^{-1}\ne 1$. In which case,
\begin{align*}\int_{G} \phi \ol{\psi}&= \int_{G} \dpn{x, \phi}{G}\dpn{x, \psi}{G}^{-1}dx \\&= \dpn{x_0, \phi}{G}\dpn{x_0, \psi}{G}\int_{G} \dpn{x - x_0, \phi}{G}\dpn{x - x_0, \psi}{G}dx \\&= \dpn{x_0, \phi}{G}\dpn{x_0, \psi}{G}\int \phi \ol{\psi}\end{align*}
so $\phi \perp \psi$.$\square$
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