Let $G$ be a compact Lie group. Let $C(G)$ denote the complex [algebra of continuous functions](/theorems/197) $G\to\mathbb C$, with pointwise addition and multiplication. For a finite-dimensional complex [vector space](/page/Vector%20Space) $V$, let $GL(V)$ denote the group of complex-linear automorphisms of $V$. A continuous finite-dimensional complex representation of $G$ means a continuous [group homomorphism](/page/Group%20Homomorphism) $\rho:G\to GL(V)$ for some finite-dimensional complex vector space $V$. For each such representation $\rho:G\to GL(V)$, each $\lambda\in V^*$, and each $v\in V$, define the matrix coefficient $c_{\lambda,v}^{\rho}:G\to\mathbb C$ by $c_{\lambda,v}^{\rho}(g)=\lambda(\rho(g)v)$. Let $\mathcal R(G)$ be the complex linear span in $C(G)$ of all such matrix coefficients. Then $\mathcal R(G)$ is a unital subalgebra of $C(G)$, is closed under pointwise complex conjugation, and separates points of $G$: for every $g,h\in G$ with $g\ne h$, there exists $f\in\mathcal R(G)$ such that $f(g)\ne f(h)$.