= Solution
Continuity immediately implies sequential continuity. Conversely, suppose a linear form $u$ is sequentially continuous but not continuous at zero. Enumerate an increasing family of seminorms that generates the <Schwartz space> topology, and let
$$
U_k=\{\varphi:p_j(\varphi)<1/k\text{ for }1\leq j\leq k\}.
$$
Since $u$ is unbounded on every neighborhood of zero, choose $\varphi_k\in U_k$ with $|u(\varphi_k)|\geq1$. For each fixed $j$, $p_j(\varphi_k)<1/k$ once $k\geq j$, so $\varphi_k\to0$ in $\mathcal S$. Sequential continuity would imply $u(\varphi_k)\to0$, a contradiction. Hence $u$ is continuous and belongs to $\mathcal S'$.
Solved by gpt-5.6-sol high.
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