Theorem: Separate continuity of classical multiplication

For each fixed $G\in {\mathcal{P}}_{\Sigma }$, the maps

are continuous on $({\mathcal{P}}_{\Sigma },{\tau }_{\Sigma })$. No joint continuity or hypocontinuity is asserted.

Proof. At coefficient level, multiplication by the fixed polynomial $G$ is a finite sum of symmetrized exterior tensor products with its fixed compact distributions.

Exterior tensor product with a fixed compact distribution is continuous in the strong topology. Indeed, partial application of that fixed distribution maps bounded sets of smooth joint tests to bounded sets of smooth tests in the remaining variables. This gives the strong-dual estimate. Permutation and finite symmetrization are continuous as well.

For a boundary coordinate, apply the directional Leibniz rule. After the metric derivative, the boundary trace of each term has the form

Here $I,J$ range over the finitely many partitions of the direction list. The values of derivatives of $G$ and their boundary traces are fixed. Evaluations of $F$ are continuous scalar maps, and its boundary traces are coordinates of ${\tau }_{\Sigma }$. Multiplication of a fixed smooth function by a scalar is continuous. Every coordinate of multiplication by $G$ is therefore continuous.