Remark 3

If there exists an integer $n\geqslant 0$ such that G is a compact subgroup of $\mathbf{G}\mathbf{L}(\mathbf{C}^n)$, the identical representation of G in $\mathbf{G}\mathbf{L}(\mathbf{C}^n)$ suffices to separate the points of G, and one can therefore then prove proposition 4 directly, and then the Peter–Weyl theorem, without appealing to the Gelfand–Raikov theorem.