Let $\mathcal{S}(\mathbb{R})$ be the Schwartz space of smooth "fast falling" functions on $\mathbb{R}$ provided with the seminorm system $\mathcal{P}=\{p_{n,m}\colon n,m\in\mathbb{N}_0\}$, where $p_{n,m}(f)=\sup_{x\in\mathbb{R}} |x^n\cdot f^{(m)}(x)|$, $f\in\mathcal{S}(\mathbb{R})$. Furthermore let the seminorm $q_{n,m}(f)=\sup_{x\in\mathbb{R}}|(1+x^2)^nf^{(m)}(x)|$ on $\mathcal{S}(\mathbb{R})$ be defined for all $n,m\in\mathbb{N}$.
Show, that the system $Q=\{q_{n,m}:n,m\in\mathbb{N}_0\}$ genereates the same topology on $\mathcal{S}(\mathbb{R})$ as $\mathcal{P}$.
I want to show, that $\tau_Q=\tau_{\mathcal{P}}$. Therefore I want to show, that $\tau_Q\subseteq \tau_\mathcal{P}$ and $\tau_\mathcal{P}\subseteq\tau_Q$, by showing that $q\in Q$ is continuos regarding $\mathcal{P}$ and vice versa $p\in\mathcal{P}$ is continuous regarding $Q$.
- $\tau_\mathcal{P}\subseteq\tau_Q$
This is clear, since $\mathcal{P}\subseteq Q$.
- $\tau_Q\subseteq\tau_\mathcal{P}$
We have some equivalent statements on how to show, that $q$ is continous regarding $\mathcal{P}$. I want to try to find $c\geq 0$ and $q_1,\dotso, q_l\in Q$ such that $q\leq c\max\{p_1,\dotso, p_l\}$
Since $f\in\mathcal{S}(\mathbb{R})$ it is $q_{n,m}(f)=\sup_{x\in\mathbb{R}}|(1+x^2)^nf^{(m)}(x)|<\infty$. Hence we can choose $c=\sup_{x\in\mathbb{R}}|x^nf^{(m)}(x)|$. Now we have to find $p\in\mathcal{P}$ such that $p_{n,m}(x)\geq 1$ for all $x\in X$ and $n,m\in\mathbb{N}$.
Can I solve this by searching for such a function? Might $e^{-x^2}+1$ do it?
Is that correct? Thanks in advance for your comments.