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 $(ii)\implies (i).$



Equivalence of different topologies of Kernel of constant coefficient hypoelliptic operator
There are at least three published proofs: The first (for the result in this generality -- for particular operators it is of course older) I know is of Malgrange
[*Existence et approximation des solutions des equations
aux derivees partielles et des equations de convolution*, Ann. Inst. Fourier, Grenoble, 6 (1955)–(1956), 271–355] using abstract results about strong
duals of Frechet-Schwartz spaces, the second in Hörmander's book *Analyiys of partial dillerential operators I* (theorem 4.4.2) uses explicitely a fundamental solution with singular support equal to the origin, and a third one is in my article *Topological properties of kernels of partial differential operators* in Rocky Mountain J. Math. 44 (2014), 1037-1052.


