I'm reading the following Theorem from Trèves book:
Theorem 24.4 Let $T$ be a distribution of order $\leq m <+\infty$ in $\Omega$, and let $S \subset \Omega$ be its support. Given any open neighbourhood $U$ of $S$ in $\Omega$, there is a family of Radon measures $\{\mu_p\}$ $(p \in\mathbb{N}^{n}, |p|\leq m)$ in $\Omega$ such that $T=\sum_{|p|\leq m}(\partial/\partial x)^p\mu_p$, and such that $\operatorname{supp} {\mu_p} \subset U$ for every $p \in \mathbb{N}^{n}$, $|p|\leq m$.
However, I got stuck in the first lines of the proof. He says:
Proof: Let $N=N(m,n)$ be the number of $n-$uples $p=(p_1,...,p_n)$ such that $|p|\leq m$. For simplicity, let us set $E_m=C_c^m(\Omega)$. There is a natural injection of $E_m$ into the product space $(E_0)^M$: it is the mapping which assigns to each $\phi \in E_m$ the set $((\partial/\partial x)^p \phi)_{(p\in \mathbb{N}^{n}, |p|\leq m}$ of its derivatives of order $\leq m$. This mapping is obviously linear, obviously not onto. But it is an isomorphism into for the structures if TVS, as immediately seen ($\phi_n$ converges to zero in $E_m$ if and only if every one of its derivatives of order $\leq m$ converges to zero in $E_0$).
The map above mentioned is given by $$\Psi:C_c^m(\Omega) \to [C_c^o(\Omega)]^N$$ given by $$\Psi(\phi)=((\partial/\partial x)^p \phi)_{p \in A},$$ where $A=\{p \in \mathbb{N}: |p|\leq m\}$ and $\# A=N$. I was able to verify linearity and injectivity. However, I was unable to justify that it is not onto continuous and $\Phi^{-1}:\operatorname{Im}(\Phi) \subset [ C_c^0(\Omega)]^{N} \to C_c^m(\Omega)$ is continuous where $\operatorname{Im}(\Phi)$ is endowed with the topology induced by $[ C_c^0(\Omega)]^{N}$.
In addition, the justification for continuity is not clear because these spaces are not metrizable (they are $LF$-spaces). The continuity of $\Phi$ I believe is valid, because if $K$ is a compact subset of $\Omega$ it is enough to prove that $\Phi|_{C_c^m(K)}:C_c^m(K) \to [C_c^0(\Omega)]^N$ is continuous. Since $\Phi$ is linear, $C_c^m(K)$ is metrizable, if $\phi_n \rightarrow 0$ in $C_c^m(K)$ then every of its derivatives of order $\leq m$ converges to zero in $C_c^0(K) \hookrightarrow C_c^0(\Omega)$. Therefore, $((\partial/\partial x)^p\phi_n(x))_{p \in A} \rightarrow 0$ in $[C_c^0(\Omega)]^N$. (Is that correct?)
My question: How to prove that $\Phi$ is not onto and $\Phi^{-1}$ is continuous?
$\Phi$ cannot be onto as the other coordinates of $\Phi(\phi)$ are already determined by its coordinate for $p = 0$. For example if $y = (y_p)_{p \in A}$ is such that $y_0 = 0$ and $y_p \neq 0$ for some $p \neq 0$, then $y \in [C_c^0(\Omega)]^N \setminus \Phi(C^m_c(\Omega))$.
The spaces $C^m_c(\Omega)$ are NOT the spaces of test functions. They are even normable by the sup norm $$||f|| = \max \{||(\partial f / \partial x)^p||_\infty; \; p \leq m \}$$ (each compactly supported function have compactly supported derrivatives).
Further, $\Phi^{-1}$ is continuous as $$\Phi(y_n) \rightarrow \Phi(y) \text{ in } \Phi(C^m_c(\Omega)) \iff \text{ for each } p \in A : \;\;(y_n)_p \rightarrow y_p \text{ in the sup norm} \iff$$ $$\iff y_n \rightarrow y \text{ in } C^m_c(\Omega).$$