# Equivalence of different topologies of Kernel of constant coefficient hypoelliptic operator

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 $$(ii)\implies (i).$$