Sequential convergence in test function space

If ${\phi }_j\to \phi$ in the LF topology, F2 places the sequence and limit in one ${\mathcal{D}}_K$. Convergence in the subspace topology follows directly: any subspace neighborhood of $\phi$ is the intersection with an ambient neighborhood, which eventually contains the sequence. By F1, $p_m({\phi }_j-\phi )\to 0$ for every $m$. Since derivatives vanish off $K$, this gives uniform convergence of every derivative on $\mathrm{\Omega }$.

Conversely suppose eventual common support and the stated uniform convergence. For $x\mathrm{\notin }K$, the eventual values ${\phi }_j(x)$ are zero, so $\phi (x)=0$; since $K$ is closed, its support is contained in $K$. The finite union $L$ of $K$ and the finitely many initial test supports is compactly inside $\mathrm{\Omega }$ and contains every support. For each $m$, uniform convergence of the finitely many derivatives through order $m$ gives $p_{m,L}({\phi }_j-\phi )\to 0$. F1 first yields convergence in ${\mathcal{D}}_L$, then LF convergence by the continuous inclusion.

These arguments also prove the stated equivalent all-support formulation. Empty $\mathrm{\Omega }$ and eventually zero sequences satisfy the same reasoning; no compactness extraction or chosen subsequence is used. The finite initial union is essential to passing from eventual to all-support language. $\mathrm{\square }$