Show that the map $(T\phi)(x) = \phi(Ax+b)$ is well-defined on $\mathcal{D}(R^n)$ and $T$ is continuous, with $A \in \mathbb{R}^{n,n},\; det(A)\neq 0$ and $b \in \mathbb{R}^n$.
Here arise the questions:
1) What does it means that $T$ is well defined on $\mathcal{D}(R^n)$? My guess is that I have to prove that $T$ is in $\mathcal{D}(R^n)$. But supposing $\phi \in \mathcal{D}(R^n)$, then $T$ is automatically in $\mathcal{D}(R^n)$ because $x \mapsto Ax+b$ is a simple translation.
2) To show that $T$ is continous I have to prove that if $\phi_k \rightarrow \phi$ in $\mathcal{D}(R^n)$, with $\{\phi_k\}$ in $\mathcal{D}(R^n)$ then it must be that $T\phi_k \rightarrow T\phi$ in $\mathcal{D}(R^n)$?