Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 327 / 1 / b / 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 letSince $u$ is unbounded on every neighborhood of zero, choose ${\phi }_k\in U_k$ with $|u({\phi }_k)|\ge 1$. For each fixed $j$, $p_j({\phi }_k)<1/k$ once $k\ge j$, so ${\phi }_k\to 0$ in $\mathcal{S}$. Sequential continuity would imply $u({\phi }_k)\to 0$, a contradiction. Hence $u$ is continuous and belongs to ${\mathcal{S}}^{\prime }$.

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