# Is this kernel space of finite dimension ?

Assume that $P \in \Psi^{m}(X)$ (X is a $C^{\infty}$ manifold)is properly supported and has a real principal part p which is homogeneous of degree m.I'm interested in the existence theorem(at least locally) for the equation $Pu=f$,according to the abstract functional analysis,we should first learn the kernel of the adjoint $P^{\ast}$,that is $$N(K)=\lbrace v \in \epsilon'(K),P^{\ast}v=0 \rbrace$$

Here K is a compact subset of X(since only locally existence is concerned)such that no complete bicharacteristic curve is contained in K.Under this condition,is N(K) a finite dimensional subspace of $C_{0}^{\infty}(K)$ ? If it is, then how to prove this ? Or are there other conditions (about the domain or the operator) to make sure the finiteness of the kernel ?

*EDIT*According to the condition on the domain,it can be shown that actually $N(K)\subset C^{\infty}(K)$,but then I don't know how to prove that the dimension is finite(i.e. the unit ball is precompact).

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If $\epsilon'(K)$ is the space $\mathscr E'(K)$ of distributions with support in $K$ and you know that $N(K)$ is contained in $C^\infty(X)$ then it is finite dimensional: Since $N(K)$ is closed in the bigger space $\mathscr E'(X)$ (the dual of the Frechet-Schwartz space $C^\infty(X)$) it is also closed in the Frechet space $C^\infty(X)$. On the other hand, closed subspaces of DFS-spaces (= strong duals of Frechet-Schwartz spaces) are again DFS and we have an open mapping theorem which yields that $N(K)$ is a Frechet space as well as a DFS-space. But such spaces are finite-dimensional (because they have a relatively compact $0$-neighbourhood).
The question when $N(K)$ is actually contained in $C^\infty(X)$ is treated in Hörmander's Propagation of Singularities and Semi-Global Existence Theorems for (Pseudo)-Differential Operators of Principal Type, Annals of Mathematics, 108 (1978),569-609.
EDIT: That Frechet-Schwartz spaces $X$ which are closed subspaces of the (strong) dual $Y'$ of a Frechet space are finite dimensional is a combination of Baire's theorem, the open mapping theorem, and a compactness argument: If $p_n$ is a sequence of semi-norms describing the topology of $Y$ one has $Y'= \bigcup_n (Y,p_n)'$ (just because continuity of a linear functional on $Y$ is continuity with respect to some $p_n$), and then Baire's theorem implies that some of the Banach spaces $Z_n=(Y,p_n)'$ (endowed with the dual norm $p_n'(\phi)=\sup \lbrace |\phi(x)|: p_n(x)\le 1\rbrace$) contains a set of second category of $X$. The open mapping theorem (as in Rudin's Functional Analysis) for $\lbrace (x,\phi)\in X\times Z_n:x=\phi \rbrace \to X$, $(x,\phi)\mapsto x$ then implies $X\subseteq Z_n$. (This is essentially the proof of Grothendieck's factrization theorem.) Since $X$ is closed even in $Y'$ (endowed with the topology of uniform convergence on bounded subsets of $Y$) it is closed in $Z_n$ and therefore, $X$ is a Banach space (the Frechet space topology coincides with the Banach topology by the open mapping theorem). But in a Frechet-Schwartz space all bounded sets (in our case, in particular the unit ball of the norm) are relatively compact, and a Hausdorff topological vector space having a relatively compact neighbourhood of zero is finite-dimensional (Rudin's book, theorem 1.22).