Corollary

The maps $\overline{\mathscr{F}}:f \rightarrow (u(f))_{u\in\widehat{G}}$ and $\mathscr{F}: A \rightarrow \sum_{u\in\widehat{G}}\langle u(g)|A_u\rangle$ induce inverse isomorphisms between the topological vector spaces $\mathscr{C}^{\infty}(G;\mathbf{C})$ and $\mathscr{S}( \widehat{G})$.

$^7$ This follows from the fact that the space $\mathscr{C}^{\infty}(G;\mathbf{C})$, with the topology of uniform

$C^{\infty}$-convergence (§6, no. 4), is complete.