Trevés, Theorem 24.4, Topological Vector Spaces, Distributions and Kernels, Dover, p. 259.
If $T$ is a distribution of finite order $\leq m$ in $\Omega$ with support $S \subset \Omega$, then given any neighbourhood $U$ of $S$ in $\Omega$, there is a family of Radon measures $\mu_p$ supported in $U$ such that $T=\sum_{|p| \leq m}(\partial/\partial x)^p \mu_p$.
The author defines $E_m=C^m_c(\Omega)$ and a map $\phi \mapsto ((\partial/\partial x)^p\phi)_{p \in \mathbb{N^n}, |p| \leq m}$ into the product $(E_0)^N$ for $\phi \in E_m$, where $N$ is the number of tuples $p=(p_1,...,p_n)$, suggests that this is an isomorphism (into), and then claims that "we may transfer any continuous linear functional on $E_m$ as a continuous linear functional on the image of this map, and then extend the latter as a continuous linear functional on $(E_0)^N$".
Could someone please elaborate on the authors claim and the map in question.
The statement is also given under the wiki-page ("structure of distributions of finite order" - in reference to Trevés):
https://en.wikipedia.org/wiki/Distribution_(mathematics)#Functions_as_distributions
The latter also appeared in connection with the following question on mathoverflow:
https://mathoverflow.net/questions/17732/difference-between-measures-and-distributions
Since $T$ has order $m$ there exists a constant $C_{U}>0$ such that $$ |T(\phi)|\leq C_{U}\sum_{|p|\leq m}\Vert(\partial/\partial x)^{p}\phi \Vert_{L^{\infty}(U)}% $$ for all $\phi\in\mathcal{D}(\Omega)$ with support in $U$. In particular, by the Hahn-Banach theorem, we can extend $T$ to a continuous linear function $$ T:C_{c}^{m}(U)\rightarrow\mathbb{R}. $$ Now consider the linear function $$ L:C_{c}^{m}(U)\rightarrow C_{c}^{0}(U,\mathbb{R}^{N}), $$ given by $$ L(\phi)=(\phi,(\partial/\partial x)^{p_{1}}\phi,\ldots,(\partial/\partial x)^{p_{N}}\phi), $$ where $p_{1}, \ldots, p_{N}$ are all the multi-indeces of order between one and $m$ (both included). Note that $L$ is injective and continuous. So $Y=L(C_{c}^{m}(U))$ is a subspace of $C_{c}^{0}(U,\mathbb{R}^{N})$ and $$ L:C_{c}^{m}(U)\rightarrow Y $$ is linear, continuous, and bijective. Given $\psi\in Y$, there exists a unique $\phi\in C_{c}^{m}(U)$ such that $L(\phi)=\psi$. Define $$ S(\psi):=T(\phi). $$ Then $S$ is linear, and $$ |S(\psi)|=|T(\phi)|\leq C_{U}\sum_{i=1}^{N}\Vert(\partial/\partial x)^{p_{i}% }\phi\Vert_{L^{\infty}(U)}=C_{U}\sum_{i=1}^{N}\Vert\psi_{i}\Vert_{L^{\infty }(U)}, $$ which implies that $S:Y\rightarrow\mathbb{R}$ is linear and continuous. By the Hahn-Banach theorem, we can extend $S$ to a linear and continuous function $S:C_{c}^{0}(U,\mathbb{R}^{N})\rightarrow\mathbb{R}$. In turn, by the Riesz representation theorem, we can find $N$ signed Radon measures $\mu_{i}$ such that $$ S(\psi)=\sum_{i=1}^{N}\int_{U}\psi_{i}\,d\mu_{i}. $$ In particular, if $\psi=L(\phi)$, then $$ T(\phi)=S(\psi)=\sum_{i=1}^{N}\int_{U}\psi_{i}\,d\mu_{i}=\sum_{i=1}^{N}% \int_{U}(\partial/\partial x)^{p_{i}}\phi\,d\mu_{i}. $$