continuous linear extension from Fréchet space $C^\infty(\Omega) $ to $C(\Omega)$ Let $\Omega\subset \mathbb{R}^n$   be an open subset, can we always extend a continuous linear functional on Fréchet space $C^\infty(\Omega)$ to a continuous linear functional on Fréchet space $C(\Omega)$?

I know if the extension exists, it must be the unique one, hence the map $(C(\Omega))' \to (C^\infty(\Omega))'$ is an injection. If the above result holds, it will an linear isomorphism, it seems not need to hold?

The usual Frèchet topology for $C^\infty(\Omega)$ is stronger than the norm topology in $C(\Omega)$. So, even though you have an inclusion as vector spaces, you don't have one as Frèchet spaces.
For a concrete example, consider the continuous linear functional $f\longmapsto f'(1)$ on $C^\infty[0,1]$. On $\mathbb R[x]\subset C[0,1]$ this map is discontinuous.
