Theorem: Separate continuity of classical multiplication

For each fixed $G\in {\mathcal{P}}_{\Sigma }$, multiplication by $G$ is continuous on $({\mathcal{P}}_{\Sigma },{\tau }_{\Sigma })$. No joint continuity or hypocontinuity is inferred.

Proof. At coefficient level, multiplication by $G$ is a finite sum of tensor products with fixed compact distributions, followed by symmetrization and degree shifts. For a fixed compact distribution $v$, partial application of $v$ maps a bounded set of smooth joint tests to a bounded set of tests in the remaining variables. This proves strong continuity of $u\mapsto u\otimes v$.

For a boundary coordinate, the differentiated Leibniz formula gives a finite sum of terms

The $G$-values and $G$-traces are fixed. The remaining maps are continuous evaluations or defining boundary coordinates. Every coordinate of multiplication by $G$ is continuous.