Lemma: Closed equation subspace and reduction

The subspace $P_m\mathcal{D}$ is closed in $\mathcal{D}$. The quotient $Q_m$ is Hausdorff. For fixed smooth backgrounds,

are continuous from tests supported in $K_f$ to ${\mathcal{D}}_C$. In particular,

when only the coordinate representatives vary.

Proof. The continuous map ${{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}-{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{R}}:\mathcal{D}\to C^{\infty }(\hat{M})$ has kernel $P_m\mathcal{D}$. If ${{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}a={{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{R}}a=u$, then $u$ is supported in the compact causal diamond of $\mathrm{supp}a$, so $u\in \mathcal{D}$ and $P_{m}u=a$. The converse follows from the Green identities.

Advanced support, compact coefficients of $L_j$, and Green continuity give the mapping assertions. Subtracting (eq:completed) gives the final identities.