the map $$\mathcal{T}_w \ni f \mapsto \pi(f)u \in \mathcal{S}_w$$ is continuous and its restriction to $\mathcal M^{\mathcal{T}_w}$ is the inverse of $V$.

Here and in the following the notation $\pi(f)u$ is motivated by the following fact.

In the framework of abstract harmonic analysis, any function $f\in L_1(G)$ defines a bounded operator $\pi(f)$ on $\mathcal{H}$, which is weakly given by $$\langle{\pi(f) v},{v'}\rangle_\mathcal{H} = \int_G f(x) \langle{\pi(x)v},{v'}\rangle_\mathcal{H} ~dx,
    \qquad v,v'\in\mathcal{H},$$ see for example Sect. 3.2 of [@fol95]. However, if $f\not\in L_1(G)$, then in general $\pi(f)v$ is well defined only if $v=u$, where $u$ is an admissible vector for the representation $\pi$.

Recalling that the (anti-)dual of $\mathcal{T}_w$ is $\mathcal{U}_w$ under the pairing , we denote by $\,^t\!{V}$ the contragradient map $\,^t\!{V}: \mathcal{U}_w\to \mathcal{S}'_w$ given by $$\langle{\,^t\!{V}\Phi },{v}\rangle_{\mathcal{S}_w}= \langle{\Phi},{Vv}\rangle_{w},\qquad
\Phi\in\mathcal{U}_w,\, v\in\mathcal{S}_w.$$

As usual, we extend the voice transform from $\mathcal{H}$ to the (anti-)dual $\mathcal{S}'_w$ of $\mathcal{S}_w$, where $\mathcal{S}'_w$ plays the role of the space of distributions. For all $T \in \mathcal{S}_w'$ we set $$V_e T(x) = \langle{T},{\pi(x)u}\rangle_{\mathcal{S}_w},
  \qquad x\in G,
\label{EVT2}$$ which is a continuous function on $G$ by item \[2.1b\]) of the previous theorem and $\langle{\cdot},{\cdot}\rangle_{\mathcal{S}_w}$ denotes the pairing between $\mathcal{S}_w$ and $\mathcal{S}'_w$, whereas $\langle{\cdot},{\cdot}\rangle_w$ is the pairing between $\mathcal{T}_w$ and $\mathcal{U}_w$.

