Let $G$ be a compact Lie group, and let $\nu:\mathcal B(G)\to[0,1]$ be the normalized Haar measure on $G$. Let $C(G)$ denote the complex [vector space](/page/Vector%20Space) of continuous functions $G\to\mathbb C$. Let $L^2(G,\nu)$ denote the complex [Hilbert space](/page/Hilbert%20Space) of $\nu$-equivalence classes of square-integrable Borel functions $G\to\mathbb C$, with [inner product](/page/Inner%20Product)
For a finite-dimensional complex vector space $W$, write $GL(W)$ for the group of invertible complex-linear maps $W\to W$. Let $\widehat G$ denote the set of isomorphism classes of irreducible finite-dimensional complex unitary representations of $G$. For each $\pi\in\widehat G$, choose a representative
paragraph
admin
\begin{align*}
\pi:G\to U(V_\pi)
\end{align*}
latex_env
admin
on a finite-dimensional complex Hilbert space $V_\pi$, and write $d_\pi:=\dim_{\mathbb C}V_\pi$.
paragraph
admin
For $\lambda\in V_\pi^*$ and $v\in V_\pi$, define the matrix coefficient
paragraph
admin
\begin{align*}
c_{\lambda,v}:G&\to\mathbb C
\end{align*}
Equip $V_\pi^*$ with the Hilbert inner product for which the basis dual to any [orthonormal basis](/page/Orthonormal%20Basis) of $V_\pi$ is orthonormal, and equip $V_\pi^*\otimes V_\pi$ with the induced tensor-product Hilbert structure. Then the map
extends by linearity and Hilbert completion to a unitary isomorphism. Equivalently, the matrix coefficient functions $\sqrt{d_\pi}\,c_{\lambda,v}$, with $\pi\in\widehat G$ and $\lambda,v$ ranging over dual orthonormal bases of $V_\pi^*$ and $V_\pi$, form an orthonormal basis of $L^2(G,\nu)$.