Space of test functions ($\mathcal{D}(U)$)

The space of test functions is $\mathcal{D}(U)=C_c^{\infty }(U)$. A sequence converges in $\mathcal{D}(U)$ when all its supports eventually lie in one compact subset of $U$ and every derivative converges uniformly there.

The space of test functions has the locally convex inductive-limit topology of the spaces of smooth functions supported in a fixed compact set $K$. Each ${\mathcal{D}}_K$ uses seminorms $p_m(\phi )={\max }_{|\alpha |\le m}\Vert {\partial }^{\alpha }\phi {\Vert }_{\infty }$. Sequential convergence means a common compact support and uniform convergence of every derivative. A linear functional on this space is a distribution exactly when its restriction to each ${\mathcal{D}}_K$ satisfies a finite order of a distribution estimate; the order and constant may depend on $K$.

Every distribution on an open set $U$ is a limit of test functions in weak convergence of distributions. On the whole space ${\mathbb{R}}^n$, choose a mollifier $\rho$ and compactly supported smooth cutoffs ${\chi }_j$ equal to one on expanding balls. Thenare test functions and $u_j\to u$: for each fixed test function $\varphi$, the cutoff is eventually one near its support, and the reflected mollifier approximates $\varphi$ with every derivative in one common compact set.

For a general open set $U$, ordinary fixed-radius convolution need not be defined near its boundary. Choose an exhaustion $K_j$ of $U$ and cutoff functions ${\chi }_j$ compactly supported inside $U$, equal to one near $K_j$. Extend the compactly supported distribution ${\chi }_{j}u$ by zero to the whole space, then convolve with a mollifier of radius ${ϵ}_j\to 0$ small enough that the resulting support stays inside $U$. For every fixed test, the cutoff eventually has no effect, so these smooth compactly supported approximants converge to $u$.