Solution
= Solution
Continuity of $u$ at zero supplies finitely many Schwartz seminorms and a constant $C$ bounding $|u[\phi]|$. Increasing their two indices to common maxima $N,k$ gives
$$
|u[\phi]|\leq C\|\phi\|_{N,k}.
$$
\b[Thus $u$ is bounded for the $X_{N,k}$ norm. Since $\mathcal S$ is dense in the Banach space $X_{N,k}$, it has a unique continuous extension $\widetilde u\in X_{N,k}'$.]