Prove that every continuous linear functional on $C^{\infty}(\Omega)$ is of the form $f$ maps to $\lambda f$ where $\lambda$ is a distribution with compact support on in $\Omega$.
Well my idea is that if we have a continuous functional on $C^{\infty}(\Omega)$, then also on $D_K$ for every compact set $K$ in $\Omega$, hence by proven result on distributions we know that since it is continuous on every $D_K$ then it is also continuous on $D(\Omega)$
HINT:
Denote by $p_{K,n}$ the seminorm $$f \mapsto p_{K,n}(f) = \sup_{x\in K, 0\le l \le n} |f^{(l)}(x)|$$
The topology of $C^{\infty}$ is given by the seminorms $p_{K,n}$, for $K$ compact in $\Omega$, $n \ge 0$. Since the map $\lambda$ is continuous there exists $K$, $n$, and $C>0$ so that $$|\lambda f| \le C\cdot p_{K,n}(f)$$
Let now $K_1$ any other compact inside $\Omega$. For $f \in D_{K_1}$ we have $$p_{K,n}(f) = p_{K_1 \cap K, n}(f) \le p_{K_1, n}(f)$$ so we get $$|\lambda f| \le C \cdot p_{K_1, n}$$ Hence, $\lambda$ is continuous on $D_{K_1}$. Therefore, $\lambda \in D'_{\Omega}$. Notice also that $\operatorname{supp} \lambda \subset K$.
The converse is a bit more delicate: Say we have a distribution with compact support $K \subset \Omega$.
Consider a conpact $K_1\subset \Omega$ so that $K \subset \overset{\circ}K_1$. We have an open cover $\Omega= \overset{\circ}K_1\cup (\Omega\backslash K)$. Let now $\psi_1$, $\psi_2$ a partition of unity corresponding to this cover, that is $\psi_i$ are smooth, $\operatorname{supp}\psi_1 \subset \overset{\circ}K_1$, $\operatorname{supp}\psi_1 \subset \Omega\backslash K$.
Let now $\phi \in D_{\Omega}$. We have $$\phi = \psi_1 \cdot \phi + \psi_2 \cdot \phi$$
Note that the support of $\psi_2 \cdot \phi$ is compact and contained in $\Omega\backslash K$. Therefore, $\lambda(\psi_2 \cdot \phi) = 0$. Now, $\psi_1 \cdot \phi \in D_{K_1}$. Therefore $$|\lambda ( \psi_1 \phi)| \le C \cdot p_{K_1, n}(\psi_1 \cdot \phi)$$
However, from Leibniz differentiation formula we get $$p_{K_1, n}(\psi_1 \cdot \phi)\le C_1 \cdot p_{K_1, n}(\phi)$$ for all $\phi$