Lemma 8

Let K be a topological field and let E and F be topological vector spaces over K. Let $f$ be a mapping of E into F and X a subset of E. Let $x\in X$. The mapping $f|X$ is continuous at $x$ if, for every neighbourhood W of 0 in F, there exist a neighbourhood U of $x$ in E and a continuous mapping $g$ of X into F such that $(f-g)(U\cap X)\subset W$.

Let $W_0$ be a neighbourhood of 0 in F and let $V_0$ be a symmetric neighbourhood of 0 in F such that $V_0+V_0+V_0\subset W_0$. Let $U_0$ be a neighbourhood of $x$ in E and $g$ a continuous mapping of X into F such that $(f-g)(U_0\cap X)\subset V_0$. There exists a neighbourhood $U_1$ of $x$ in E such that $g(U_1\cap X)\subset g(x) + V_0$. For every $y\in U_0\cap U_1\cap X$, one has

hence $f(y)\in f(x) + W_0$, whence the result.

Let us now prove theorem 3. Let us denote by Pos$_1(G)_f$ (resp. Pos$_1(G)_c$) the set Pos$_1(G)$ endowed with the weak topology (resp. with the topology of compact convergence); analogously, let us denote by $\mathscr{C}_b(G)_f$ (resp. $\mathscr{C}_b(G)_c$) the space $\mathscr{C}_b(G)$ endowed with the weak topology (resp. with the topology of compact convergence).

The identity mapping of Pos$_1(G)_c$ into Pos$_1(G)_f$ is continuous (lemma 5). Conversely, let $\iota$ denote the inclusion of Pos$_1(G)_f$ into $\mathscr{C}_b(G)_c$. Let us prove that $\iota$ is continuous, to conclude the proof.

Let $\varphi_1\in$ Pos$_1(G)$. To verify that $\iota$ is continuous at $\varphi_1$, we apply lemma 8. Let W be a neighbourhood of 0 in $\mathscr{C}_b(G)_c$. It is enough to find a linear mapping $u$ of $\mathscr{C}_b(G)$ into $\mathscr{C}_b(G)$ which induces, by passing to subspaces, a continuous mapping of Pos$_1(G)_f$ into $\mathscr{C}_b(G)_c$, together with a neighbourhood U of 0 in Pos$_1(G)_f$, such that one has $(\iota -u)(U\cap$ Pos$_1(G))\subset W$.

There exists $\varepsilon >0$ such that W contains the set of $\varphi \in \mathscr{C}_b(G)$ satisfying $\|\varphi \|_{\infty}\leqslant \varepsilon$. Since $\varphi_1(e) = 1$, there exists a compact neighbourhood V of $e$ in G such that

sup$_{x\in V}|1-\varphi_1(x)|\leqslant \frac{\varepsilon^2}{4}$.

Let $\varphi_V$ be the characteristic function of V, and put $\psi_V=\mu(V)^{-1}\varphi_V$. The linear mapping $u:\varphi \mapsto \psi_V*\varphi$ of Pos$_1(G)_f$ into $\mathscr{C}_b(G)_c$ is continuous (lemma 6).

Let $q_V$ denote the seminorm $\varphi \mapsto |\langle \varphi , \psi_V\rangle |$ on $\mathscr{C}_b(G)$; it is continuous for the weak topology. Since $\psi_V$ vanishes outside V, we have $q_V(1-\varphi_1)\leqslant \varepsilon^2/4$; there therefore exists a neighbourhood U of $\varphi_1$ in $\mathscr{C}_b(G)_f$ such that $q_V(1-\varphi )\leqslant \varepsilon^2/2$ for every $\varphi \in U$.

Let $\varphi \in U\cap$ Pos$_1(G)$. By lemma 7, one has

therefore $(\iota -u)(U)\subset W$. The theorem is proved.

In general, the weak topology on Pos$_{\leqslant 1}(G)$ does not coincide with the topology of compact convergence, as is shown by the example of the group $\mathbf{R}$.

We recall that $\widehat{G}$ denotes the set of classes of irreducible unitary representations of G (V, p. 393, def. 8).