Let $P(D)$ be hypoelliptic operator with constant coefficients in $\mathbb{R^n}$.Let $\Omega$ be an open subset of $\mathbb{R^n}$ and $\mathscr{N_\Omega}$ denote space of distribution solutions of homogeneous equation $P(D)h=0 $. I need to prove that following topologies on $\mathscr{N_\Omega}$ are identical:
(i) the $C^{\infty}$ topology (the uniform convergence of the functions and all their derivatives on every compact subset of $\Omega$),
(ii) the $C^0$ topology(the uniform convergence of the functions on every compact subset of $\Omega$),
(iii) the topology induced by $\mathscr{D'}(\Omega)$(the functions $f_{\alpha}\in \mathscr{N_\Omega}$ converge if the integrals $\int f_{\alpha} \phi dx$ converge for every test function $\phi \in \mathscr{D}(\Omega)$, uniformly on bounded subsets of $\mathscr{D}(\Omega)$ ).
We know that $\mathscr{N_\Omega} \in C^{\infty}(\Omega)$. Clearly $(i)\implies (ii) \implies (iii)$.
How do I see other way implications perticularly $(ii)\implies (i).$
