> [!definition] > > Let $U \subset \real^d$ be [[Open Set|open]], then the set $\ce'(U)$ is the space of [[Compactness|compactly]] [[Support of Distribution|supported]] [[Distribution|distributions]]. > [!theorem] > > Let $T \in \ce'(U)$, $\phi \in \cd(U)$, and > $ > V = \bracs{x \in \real^d: x - \supp{\phi} \subset U} > $ > be the domain of the [[Convolution of Distribution|convolution]]. If $U + \supp{\psi} \subset V$, then $T * \phi \in C_c^\infty(V)$. In particular, if $U = \real^d$, then $V = \real^d$ and $T * \psi$ is always compactly supported. > > *Proof*. Let $x \in \real^d$ such that $x \not\in \supp T + \supp \psi$, then > $ > (F * \phi)(x) = \anglesn{F, \tau_{x}\td \psi} = 0 > $ > since $\text{supp}(\tau_x \td \psi) \cap \supp{T} = \emptyset$. Therefore > $ > \supp{T * \psi} \subset \supp T + \supp \psi \subset U + \supp{\psi} > $ > [!theorem] > > Let $T \in \ce'(U)$, then there exists $\seq{\psi_n} \subset C_c^\infty(U)$ such that $\psi_n \to T$ in $\cd'(U)$. > > *Proof*. Let $\phi \in \cd$ be a [[Mollifier|mollifier]], then for sufficiently small $t$, $\phi_t * T \in C_c^\infty(U)$. Sending $t \to 0$ yields the desired result. > [!theorem] > > Let $F \in \ce'(U)$, then $F$ extends uniquely to a continuous linear functional on $C^\infty(U)$. Let $G \in C^\infty(U)^*$, then $G|C_c^\infty(U) \in \ce'(U)$. > > *Proof*. Let $F \in \ce'(U)$. Let $\psi \in C_c^\infty(U, [0, 1])$ with $\psi|_{\supp{F}} = 1$. Define the extension > $ > \ol{F}: C^\infty (U) \to \complex \quad \anglesn{\ol F, \phi} = \angles{F, \psi \phi} > $ > since $\supp{\psi}$ is a compact neighbourhood of $\supp{F}$, we can restrict $F$ to $\cd(\supp{\psi})$. In particular, the topology on $\cd(\supp{\psi})$ is defined by a family of seminorms. The continuity of $F|_{\cd(\supp{\psi})}$ implies the existence of $N \in \nat$ and $C \ge 0$ such that > $ > |\anglesn{\ol F, \phi}| \le C\sum_{\abs{\alpha} \le N}\norm{\partial^\alpha(\psi \phi)}_{u, \supp{\psi}} \le C' \sum_{\abs{\alpha} \le N}\norm{\partial^\alpha \phi}_{u, U} > $ > therefore $\ol F \in C^\infty(U)^*$. > > Now suppose that $G \in C^\infty(U)^*$, then there exists $K \subset U$ [[Compactness|compact]], $N \in \nat$, and $C \ge 0$ such that > $ > |\anglesn{G, \phi}| \le C\sum_{\abs{\alpha} \le N}\norm{\partial^\alpha \phi}_{u, K} > $ > and $K$ contains our support. If $\seq{f_n} \subset \cd(U)$ are supported by a common compact set $K'$ and converge to $f$ in $\cd(U)$, then > $ > \begin{align*} > |\anglesn{G, f_n - f}| &\le C\sum_{\abs{\alpha} \le N}\norm{\partial^\alpha (f_n - f)}_{u, K} \\ > &= C\sum_{\abs{\alpha} \le N}\norm{\partial^\alpha (f_n - f)}_{u, K \cap K'} \\ > &\le C\sum_{\abs{\alpha} \le N}\norm{\partial^\alpha (f_n - f)}_{u, K'} \to 0 > \end{align*} > $ > as $n \to \infty$. So $G \in \cd(U)$ with $\supp{G} \subset K$.