I am studying distribution theory, specifically the Schwarz space of rapidly decaying functions and its dual, the space of tempered distributions. One of my my problem involves proving that
The map $(1+x^2)^v \times:\phi(x) \rightarrow (1+x^2)^v\phi(x)$ is a linear and continuous map on $S(\mathbb{R})$ and $S'(\mathbb{R})$.
In order to do this, I need to prove that
i) $(1+x^2)^v\phi(x) \in S(\mathbb{R}), (S'(\mathbb{R}), \text{respectively})$ (done due to the fact that the derivative of $(1+x^2)^v$ is just a polynomial)
ii) $(1+x^2)^v \times$ is linear (clear)
iii) $(1+x^2)^v \times$ is continuous (seems intuitive but not so clear to me)
How do I prove that a map on $S(\mathbb{R})$ (and $S'(\mathbb{R})$) is continuous? I know that $C^\infty(\Omega)$ is a Frechet space, equipped with a series of semi-norms given by $f_n = \Sigma_{\alpha \leq n} \sup_{x\in K_n}|\partial^\alpha u(x)|$ for $u\in C^\infty(\Omega)$, and the equipped metric distance is $d(u,v) = \Sigma_{n=0}^{+\infty}2^{-n}\frac{f_n}{1+f_n}$.
I have tried to use the definitions of continuous map between topological spaces, i.e, the pre-image of an open set is open but it does not seem possible.
Can you help me with this exercise please?