Theorem 1

a) Let $f$ be an infinitely-differentiable complex function on G. Then the family $(f_u)_{u\in\widehat{G}}$, where $f_u(g) =\langle u(g)|u(f)\rangle$, is uniformly summable on G and, for all $h\in G$,

$f(h) =\sum_{u\in\widehat{G}}\langle u(h)|u(f)\rangle =\sum_{u\in\widehat{G}}d(u)\int_Gf(g$)Tr($u(gh^{-1})$)$dg$.

b) Let $f$ be an integrable function on G; then $f$ is equal almost everywhere to an infinitely-differentiable function if and only if the function $u \rightarrow \|u(f)\|_{\infty}$ is rapidly decreasing on $\widehat{G}$.

Let $f$ be an infinitely-differentiable function on G, and let $\Gamma$ be a Casimir element for G; by formula (14),

for all $n\geq 0$, and consequently

$\widetilde{\Gamma}(u)^n\|u(f)\|_{\infty}\leq \|(L_{\Gamma})^nf\|_1\leq$ sup$_{g\in G}|((L_{\Gamma})^nf)(g)|$; (15)

thus, the function $u \rightarrow \|u(f)\|_{\infty}$ is indeed rapidly decreasing.

Conversely, let $A = (A_u)_{u\in\widehat{G}}$ be an element of $F( \widehat{G})$ such that the function $u \rightarrow \|A_u\|_{\infty}$ is rapidly decreasing. Put $f_u(g) =\langle u(g)|A_u\rangle$; the function $g \rightarrow f_u(g)$ is analytic, hence infinitely-differentiable. By Chap. III, §3, no. 7, Prop. 27,

for all $x\in \mathfrak{g}$. Let $t\in U(G)$; by the preceding formula,

and consequently

Since $d(u)$ and $\|u(t)\|_{\infty}$ are moderately increasing (Prop. 3) and $\|A_u\|_{\infty}$ is rapidly decreasing, the function $u \rightarrow$ sup$_g|(L_tf_u)(g)|$ is rapidly decreasing; thus, the family $(L_tf_u)_{u\in\widehat{G}}$ is uniformly summable. It follows$^7$ that the sum of the family $(f_u)$ is an infinitely-differentiable function on G, whose Fourier cotransform is $(A_u)$, hence the theorem.

Denote by $\mathscr{S}( \widehat{G})$ the vector subspace of $L^2( \widehat{G})$ consisting of the families $A = (A_u)_{u\in\widehat{G}}$ such that the function $u \rightarrow \|A_u\|_{\infty}$ is rapidly decreasing on $\widehat{G}$. It follows from the theorem that the maps $\mathscr{F}:f \rightarrow (u(f))_{u\in\widehat{G}}$ and $\mathscr{F}: A \rightarrow \sum_{u\in\widehat{G}}\langle u(g)|A_u\rangle$ induce inverse isomorphisms between the complex vector spaces $\mathscr{C}^{\infty}(G;\mathbf{C})$ and $\mathscr{S}( \widehat{G})$. Give the space $\mathscr{C}^{\infty}(G;\mathbf{C})$ the topology of uniform $C^{\infty}$-convergence (§6, no. 4) which can be defined by the family of semi-norms $f \rightarrow$ sup$_{g\in G}|L_tf(g)|$ for $t\in U(G)$, and the space $\mathscr{S}( \widehat{G})$ the topology defined by the sequence of semi-norms $p_n: A \rightarrow$ sup$_{u\in\widehat{G}}( \widetilde{\Gamma}(u) + 1)^n\|A_u\|_{\infty}$. Formula (15) of the preceding proof shows that $\overline{\mathscr{F}}$ is continuous. Let $t\in U(G)$, and let $A = (A_u)_{u\in\widehat{G}}$ be an element of $\mathscr{S}( \widehat{G})$; put $f_n(g) =\langle u(g)|A_u\rangle$. Let $p$ be an integer such that $\sum_{u\in\widehat{G}}\widetilde{\Gamma}(u)^{-p}= M<\infty$. By the preceding proof, there exists a positive integer $m$ such that, for all $g\in G$,

so $|(L_t\mathscr{F}(A))(g)| \leq mMp_m(A)$; this proves that $\mathscr{F}$ is continuous. Consequently: