Test function lf topology universal property

Finite intersections of translated seminorm balls form a topology: each ball is stable under sufficiently small translates, by $\mathrm{\mid }q(f)-q(g)\mathrm{\mid }\le q(f-g)$, and intersections refine intersections. Addition is continuous since $q(h+k)\le q(h)+q(k)$. Joint scalar multiplication is continuous at $(a,f)$ since $q(bg-af)\le \mathrm{\mid }b\mathrm{\mid }q(g-f)+\mathrm{\mid }b-a\mathrm{\mid }q(f)\mathrm{.}$ Near $a$, bound $\mathrm{\mid }b\mathrm{\mid }$ by $\mathrm{\mid }a\mathrm{\mid }+1$ and make the two terms small for each of finitely many seminorms. Balls at zero are convex and balanced by the seminorm inequalities. Thus the construction is a locally convex vector topology.

Each inclusion is continuous by the definition of $\mathcal{P}$. Define $Q_m(f)={\max }_{\mathrm{\mid }\alpha \mathrm{\mid }\le m}{\sup }_{x\in \mathrm{\Omega }}\mathrm{\mid }{\mathrm{\partial }}^{\alpha }f(x)\mathrm{\mid }$, with value zero on the empty domain. Compact support makes this finite and its restriction to ${\mathcal{D}}_K$ is $p_m$, so $Q_m\in \mathcal{P}$. In particular $Q_0$ separates points: if $f\mathrm{\ne }g$, the disjoint balls of radius $Q_0(f-g)\mathrm{/}3$ about them separate them. On ${\mathcal{D}}_K$ the induced topology is no finer than its prescribed topology by inclusion continuity, and no coarser because all $p_m$ are restrictions of $Q_m$.

Let $L$ be linear. If $L$ is continuous, composing with each continuous inclusion proves continuity of every restriction. Conversely suppose all restrictions are continuous. For each seminorm $r$ generating the topology of $E$, $r\circ L$ is a seminorm whose fixed-support restrictions are continuous. Thus $r\circ L\in \mathcal{P}$, and inverse images of translated finite seminorm balls are open, proving continuity of $L$. Applying this result to the identity into any other locally convex topology with continuous inclusions shows that topology is contained in the constructed one.

Let $(K_j)$ be increasing compact sets with interiors covering $\mathrm{\Omega }$. Every compact $K\subseteq \mathrm{\Omega }$ is contained in some $K_N$, by a finite subcover of these interiors and the maximum of its indices. The inclusion ${\mathcal{D}}_K\to {\mathcal{D}}_{K_N}$ is continuous since every derivative seminorm restricts to the same seminorm. Hence a seminorm continuous on all ${\mathcal{D}}_{K_j}$ is continuous on every ${\mathcal{D}}_K$, and the converse is immediate. The defining family of seminorms is therefore identical for any such exhaustion; step 3.1 also gives the stated map criterion using only exhaustion stages. By F2 these stages are Fréchet spaces. For $\mathrm{\Omega }=\mathrm{\varnothing }$ they and the limit are zero, and the same statements hold. No witnesses are selected for infinitely many compact sets: the containment argument is for one fixed $K$ at a time. $\mathrm{\square }$