and only if $\lim_{n\to+\infty} \lVert f_n \rVert_{L_{p,w}}=0$ for all $1<p<\infty$. With this topology, $\mathcal{T}_w$ becomes a reflexive Frechét space. The (anti)-linear dual space of $\mathcal{T}_w$ can be identified with $$\mathcal{U}_w = {\operatorname{span}}\bigcup_{1<q<\infty} L_{q,{w^{-1}}}(G)$$ under the pairing $$\int_G \Phi(x)\overline{f(x)}~dx=\langle{\Phi},{f}\rangle_w,\qquad
  \Phi \in \mathcal{U}_w,\,f\in\mathcal{T}_w.\label{eq:67}$$

The space $\mathcal{U}_w$ is endowed with one of the following equivalent topologies, both compatible with the pairing .

i)  The finest topology making the inclusions $L_{q,{w^{-1}}}(G)\hookrightarrow \mathcal{U}_w$ continuous for all $1<q<\infty$.

ii) The topology induced by the family of semi-norms $\left(\lVert{\cdot}\rVert_{p,r}\right)_{1 < p < r < \infty}$, where $$\lVert{\Phi}\rVert_{p,r}
               = \sup
                   \left\{
                         \lvert{\langle{\Phi},{f}\rangle_w}\rvert
                         ~\middle|~
                         f \in \mathcal{T}_w
                         \text{ and }
                         \max\left\{\lVert{f}\rVert_{L_{p,w}}, \lVert{f}\rVert_{L_{r,w}} \right\} \leq 1
                       \right\},$$ for $\Phi \in \mathcal{U}_w$.

The representation $\lambda$ leaves invariant both $\mathcal{T}_w$ and $\mathcal{U}_w$, it acts continuously on $\mathcal{T}_w$, and the contragradient representation $\,^t\!{\lambda}$ of $\lambda_{|\mathcal{T}_w}$, given by $$\langle{\,^t\!{\lambda}_g \Phi},{f}\rangle_w =\langle{ \Phi},{\lambda_{g^{-1}} f}\rangle_w \quad \text{for} \quad \Phi \in \mathcal{U}_w
      \quad \text{and} \quad f \in \mathcal{T}_w,$$ is simply $\,^t\!{\lambda} = \lambda_{|\mathcal{U}_w}$.

Take $g \in \mathcal{T}_w$ with $\widecheck{g} \in \mathcal{T}_w$. For all $f \in \mathcal{T}_w$ the convolution $f \ast g$ is in $\mathcal{T}_w$ and the map $$f\mapsto f\ast g$$ is continuous from $\mathcal{T}_w$ into $\mathcal{T}_w$. Furthermore, for all $\Phi \in \mathcal{U}_w$ the convolution $\Phi \ast g$ 