Proposition: Hausdorffness and basic continuity

The topology ${\tau }_{\Sigma }$ is Hausdorff. Complex conjugation and every fixed-direction operator ${\mathrm{D}}_{\eta }:{\mathcal{P}}_{\Sigma }\to {\mathcal{P}}_{\Sigma }$ are continuous. Evaluation of a polynomial or any of its fixed directional derivatives at a fixed smooth configuration is continuous.

Proof. The coefficient inclusion is injective by Lemma lem:kernelunique and takes values in a Hausdorff space. Hence the graph topology is Hausdorff.

Evaluation and directional contraction with fixed smooth fields are continuous on each strong compact-distribution coefficient space. To see this directly, a bounded set of smooth target tests is mapped to a bounded set of smooth source tests by multiplication with the fixed field or by insertion in a kernel slot. Strong-dual seminorms therefore give the required bounds. The degree shifts define continuous maps on the locally convex direct sum.

For boundary coordinates,

Thus differentiation shifts the already included coordinates. Conjugation sends the coordinate at $(\Psi ,\mathbf{\eta })$ to the conjugate coordinate at $(\overline{\Psi },\overline{\mathbf{\eta }})$. These operations are continuous in the smooth target spaces.