Want to show that the topology on $C_{\mathrm{compact}}^{\infty}(R)$, which is given by all the good semi-norms, is generated by the following collection of semi-norms $\| \cdot\|_{m,\epsilon}$
$m=\{m_i\}_{i=1}^{\infty}$ is an increasing positive integer sequence,
$\epsilon=\{\epsilon_i\}_{i=1}^{\infty}$ is a sequence converging to zero from above.
$K_{i}=[-i,+i],\quad K_{0}=\phi$ ,
$\|\psi\|_{m,\epsilon}=\sup_{n\ge0}\epsilon_{n}^{-1}\sup_{x\notin K_{n}}\sum_{m_{n}\ge j\ge0}|D^{j}\psi(x)|$
I proved that $||\cdot||_{m,\epsilon}$ is a good semi-norm. How to show that every good semi-norm can be cast in to this form? Is this statement even true? Thanks.
good seminorm means it is continous in $C_{\mathrm{compact}}^{\infty}(K)$ for all compact $K$
Topology in $C_{\mathrm{compact}}^{\infty}(K)$ is induced by $\{\| \cdot\|_{C^{0}},...,\| \cdot\|_{C^{n}},...\}$ $\|\psi\|_{C^{k}}=\sup_{x\in K}\sum_{k\ge j\ge0}|D^{j}\psi(x)|$