Let $n , m \in \mathbb { N }$ and let $r , s \in \mathbb { R } _ { + } ^ { * } , r < \rho$. 8a. Show that $\| \cdot \| _ { r , s }$ is a norm on $\mathscr { D } _ { \rho } \left( \mathbb { R } _ { n } [ X ] \right)$. 8b. Show that if $P \in \mathscr { D } _ { \rho } \left( \mathbb { R } _ { n } [ X ] \right)$ and $Q \in \mathscr { D } _ { \rho } \left( \mathbb { R } _ { m } [ X ] \right)$, then $P Q \in \mathscr { D } _ { \rho } \left( \mathbb { R } _ { n + m } [ X ] \right)$ and $$\| P Q \| _ { r , s } \leqslant \| P \| _ { r , s } \cdot \| Q \| _ { r , s }$$
Let $n , m \in \mathbb { N }$ and let $r , s \in \mathbb { R } _ { + } ^ { * } , r < \rho$.
\textbf{8a.} Show that $\| \cdot \| _ { r , s }$ is a norm on $\mathscr { D } _ { \rho } \left( \mathbb { R } _ { n } [ X ] \right)$.
\textbf{8b.} Show that if $P \in \mathscr { D } _ { \rho } \left( \mathbb { R } _ { n } [ X ] \right)$ and $Q \in \mathscr { D } _ { \rho } \left( \mathbb { R } _ { m } [ X ] \right)$, then $P Q \in \mathscr { D } _ { \rho } \left( \mathbb { R } _ { n + m } [ X ] \right)$ and
$$\| P Q \| _ { r , s } \leqslant \| P \| _ { r , s } \cdot \| Q \| _ { r , s }$$