Identification of $\mathcal E’(U)$.

Question

We define $\mathcal{E}(U)$ as the space $C^\infty(U)$ ($U$ is open in $\mathbb{R}^n$) with topology induced by seminorms $$p_{K,\alpha}(\psi) =\sup_{K}|\partial^\alpha \psi |, K\subset\subset U, \alpha\in \mathbb{N}^n. $$ $\mathcal{E}’(U)$ is the continuous linear functional of $\mathcal{E}’(U)$. In Hömander’s book, $\mathcal E’(U)$ is identical with distributions(in $\mathcal D’(U)$) with compact support. The book gives details how to construct the extension of $u\in \mathcal D’$ to $\bar u\in \mathcal E ’$, by let $$(\bar u, \psi )=(u, \phi\psi ),\forall \psi\in C^\infty(U)$$ where $\phi$ is some test function. But I’m confused with the another side: Is $u\in \mathcal E’$ restricted to $C_c^\infty(U)$ a distribution on $U$ with compact support? how is the compact set chosen? And how does the bounded condition hold?

Answer 1

From the definition of the topology of $\mathcal{E}(U)$ there exists a compact $K$, an integrer $m$ and a constant $C>0$ such that $$|\langle u,\varphi\rangle|\leq C \sup_{x \in K} \sup_{|p|\leq m} |(\partial/\partial x)^p\varphi(x)| \quad \hbox{ for all } \varphi \in C^\infty(U).$$ In particular, if $\varphi \in C_0^\infty(U)$ and $\operatorname{supp} \varphi \subset U \setminus K$ we obtain $|\langle u, \varphi \rangle|\leq 0$, that is, $\langle u, \varphi \rangle=0$. This means that $u$ vanishes in $U \setminus K$. Therefore, $\operatorname{supp} u \subset K$.

Thus, the compact $K$ comes from the definition of the topology of $\mathcal{E}(U)$.