Corollary 1

The vector space $\Theta (G^{\sharp})$ is dense in $\mathscr{C}(G^{\sharp})$ and in $L^2(G^{\sharp})$.

The second assertion follows from prop. 9. As to the first assertion, consider the linear representation $\varrho$ of G on the Banach space $\mathscr{C}(G)$ defined by

for every $f\in \mathscr{C}(G)$ and every $x\in G$. It is a continuous isometric representation, and $\mathscr{C}(G^{\sharp})$ is the space of the elements invariant under this representation. The continuous projector $p=\varrho (1)$ of $\mathscr{C}(G)$ therefore has image $\mathscr{C}(G^{\sharp})$ (cor. 3 of V, p. 466); it has norm $\leqslant 1$.

Let $f\in \mathscr{C}(G^{\sharp})$ and let $\varepsilon >0$. There exists $\widetilde{f}\in \Theta (G)$ such that $\|f-\widetilde{f}\|\leqslant \varepsilon$ (prop. 4 of V, p. 462), and then $\|f-p(\widetilde{f})\|=\|p(f-\widetilde{f})\|\leqslant \varepsilon$; since $p(\widetilde{f})\in \mathscr{C}(G^{\sharp})$, we conclude that $\Theta (G^{\sharp})$ is dense in $\mathscr{C}(G^{\sharp})$.

Let $R(G)$ denote the Grothendieck ring of continuous representations of $G$ in finite-dimensional separated complex vector spaces (cf. LIE, IX, p. 70 and A, VIII, p. 182, applied to the additive class of continuous representations of $G$ in finite-dimensional separated complex vector spaces, viewed as $\mathbf{C}$[G]-modules). Since every continuous linear representation of $G$ in a finite-dimensional separated topological vector space is semisimple (prop. 1 of V, p. 457), the abelian group $R(G)$ is free and the classes of irreducible unitary representations $\pi \in \widehat{G}$ form a basis of it (A, VIII, p. 186, prop. 7).