Proposition: Operator Dini interface

Let continuous linear $G_{\delta }:P\to {\mathcal{D}}^{\prime }(X;E)$ have scalar scale regularity and annihilate $\ker \mathsf{Rec}$. Suppose

and

for continuous input seminorms $p_H,q_H$. Then the limit is continuous and linear, uniform on bounded input and test sets, and descends continuously to $K$.

Proof. The fixed-kernel integration estimate carries the factor $p_H(A)$. Together with the anchor it bounds every strong output seminorm by a continuous input seminorm. Continuous seminorms are bounded on bounded input sets. Null annihilation passes to the limit.