Test function topology

Let $\mathrm{\Omega }\subseteq {\mathbb{R}}^n$ be open. For the fixed-support spaces of Fixed support test function frechet space, let $\mathcal{P}$ be the set of all seminorms $q:\mathcal{D}(\mathrm{\Omega })\to [0,\mathrm{\infty })$ such that $q{\mathrm{\mid }}_{{\mathcal{D}}_K}$ is continuous for every compact $K\subseteq \mathrm{\Omega }$. A seminorm means $q(av)=\mathrm{\mid }a\mathrm{\mid }q(v)$ and $q(v+w)\le q(v)+q(w)$.

The test-function topology is the topology generated by translated finite intersections of sets $\{\phi :q(\phi )<\epsilon \}$, where $q\in \mathcal{P}$ and $\epsilon >0$. Equivalently it is the finest locally convex topology making every inclusion ${\mathcal{D}}_K\hookrightarrow \mathcal{D}(\mathrm{\Omega })$ continuous; the universal-property lemma below proves this equivalence and the topology assertions. This is the locally convex inductive-limit or LF topology, not the unrestricted final topology of arbitrary spaces.

For precision, a locally convex topology here is a vector topology generated by a family of seminorms, and boundedness means absorption by every zero-neighborhood: for each such neighborhood $U$ there is $t>0$ with $B\subseteq tU$. Finite intersections may be empty, giving the whole space. When $\mathrm{\Omega }$ is empty there is only the zero vector and its unique topology. No convergence criterion for sequences is used in the definition; that criterion will be a theorem.