= Test-function inductive limit topology
{title2=$\mathcal D(X)=\varinjlim_K\mathcal D_K(X)$}
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> $p_m(\varphi)=\max_{|\alpha|\leq m}\|\partial^\alpha\varphi\|_\infty$. 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 $K$.
Back to article page
