> [!definition] > > Let $U \subset \real^d$ be an [[Open Set|open set]]. For each $K \subset U$ [[Compactness|compact]], let > $ > \cd_K(U) = \bracs{\phi \in C_c^\infty(U): \supp{\phi} \subset K} > $ > be the space of [[Space of Smooth Functions|smooth functions]] on $U$ with [[Support of Function|support]] in $K$, equipped with the [[Seminorm|seminorms]] $\bracsn{\norm{ \cdot }_{K, \alpha}: \alpha \in \nat_0^d}$, where $\norm{\phi}_{K, \alpha} = \norm{\partial^\alpha \phi|_K}$, then each $\cd_K(U)$ is a [[Fréchet Space|Fréchet space]]. > > Let $I$ be the collection of all compact subsets of $U$, ordered by inclusion, then $(\bracs{\cd_K(U)}, \bracsn{\iota^K_L: K \subset L})$ forms a strict inductive system of [[Locally Convex Topological Vector Space|locally convex topological vector spaces]]. Their direct limit > $ > \cd(U) = \varinjlim \cd_K(U) > $ > is the space of **test functions** on $U$, consisting of all [[Compactly Supported|compactly supported]] smooth functions on $U$, equipped with the final locally convex topology. > [!theorem] > > Let $U \subset \real^d$ be open, then $\cd(U)$ is a [[LF-Space]], and > 1. A [[Sequence|sequence]] $\seq{\phi_n} \subset \cd(U)$ converges to $\phi \in \cd(U)$ if and only if there exists $K \subset U$ compact such that $\supp{\phi_n} \subset K$ for all $n \in \nat$ and $\phi_n \to \phi$ uniformly along with all of their derivatives. > 2. $\cd(U)$ is sequentially complete. > 3. If $F$ is a locally convex topological vector space, and $T: E \to F$ is linear, then $T$ is continuous if and only if $T$ is sequentially continuous. > > *Proof*. Since open subsets of $\real^d$ are $\sigma$-compact, there exists a countable cofinal subset of the family of compact sets on $U$, which makes the limit countable.