Let K be a compact set. How does one show the following?
If a linear map $T:C^\infty_c(K) \to X$ into a normed vector space X is continuous then there exists $k \geq 0$ and $C>0$ such that $\|Tf\|_X \leq C\|f\|_{C^k}$ for all $f \in C^\infty_c(K)$.
$C^\infty_c(K)$ is not a normed vector space, correct? What notions of continuity hold for a map $C^\infty_c(K) \to X$, just the notion for a map between two topological spaces?