Extending a distribution continuously to $C_c^N (\Omega)$

Question

Let $\Omega \subseteq \mathbb{R}^n$ be a domain and let $u\in D'(\Omega)$ be a distribution of order $\leq N$.

How can we show that $u$ can be continuously extended to $C_c^N(\Omega)$?

By "continuously extended" I mean: extending $u$ to a linear functional on $C_c^N(\Omega)$ such that if $f_n, f$ have the same compact support and $f_n \rightarrow f$ in $C^N$, then $u(f_n)\rightarrow u(f)$

Answer 1

First prove

for any $\phi\in C^{N}_c(\Omega)$, we can find a sequence $\phi_v\in C^{\infty}_c(\Omega)$ supported in fixed compact neighborhood $K$ of support of $\phi$ such that

$$\sum_{|\alpha|\le N}\sup|\partial^{\alpha}\phi - \partial^{\alpha}\phi_v|\to 0,\quad v\to \infty$$

And define $u(\phi) = \lim u(\phi_v)$. The limit exists since

$|u(\phi_v) - u(\phi_w)| = |u(\phi_v-\phi_w)|\le C\sum_{|\alpha|\le N}\sup|\partial^{\alpha}(\phi_v-\phi_w)|\to 0$.