Fixed support test function frechet space

For an integer $n\ge 1$, a compact $K\subseteq \mathrm{\Omega }$, and an open set $\mathrm{\Omega }\subseteq {\mathbb{R}}^n$, put ${\mathcal{D}}_K=\{\phi \in \mathcal{D}(\mathrm{\Omega }):\mathrm{supp}\phi \subseteq K\}$, using Test function space d of an open set. For integers $m\ge 0$ set

$p_m(\phi )={\max }_{\mathrm{\mid }\alpha \mathrm{\mid }\le m}{\sup }_{x\in K}\mathrm{\mid }{\mathrm{\partial }}^{\alpha }\phi (x)\mathrm{\mid },d(\phi ,\psi )={\sum }_{m=0}^{\mathrm{\infty }}{2}^{-m-1}\min (1,p_m(\phi -\psi ))\mathrm{.}$

The topology is generated by the seminorms $p_m$: neighborhoods at zero contain finite intersections of $p_m<\epsilon$ with $\epsilon >0$. Derivatives are continuous on $K$, so these suprema are finite; for $K=\mathrm{\varnothing }$ all suprema are defined as zero and ${\mathcal{D}}_K=\{0\}$. For nonempty $K$, $p_0$ separates functions because all functions vanish off $K$. These are increasing seminorms and the displayed series converges since its terms are at most $2^{-m-1}$.

The term fixed-support Fréchet space refers to this topology and metric. Their agreement and completeness are established by the registered justifier Fixed support test function spaces are complete ↗, rather than included as unproved consequences of the name. If $K$ has empty interior, every test supported in $K$ is zero: a nonzero continuous value would have a neighborhood of nonzero values inside $K$.