Theorem: No normal-topology extension of the mandatory coefficient

Choose $f,f^{\prime },g$ as in Lemma lem:separated, and let $K$ be a compact set containing neighborhoods of both mixed-kernel supports. The map $T_g$ has no extension to a domain containing

which:

1. agrees with $T_g$ on the regular kernels;

2. is sequentially continuous at $(b_f,b_{f^{\prime }})$ for the product of the subspace normal topologies inherited from ${{\mathcal{E}}^{\prime }}_{{\Gamma }_B}(K)$.

Proof. By Lemma lem:normalconv,

in the stated product topology. Sequential continuity would force $T_g(b_{f,\epsilon },b_{f^{\prime },\epsilon })$ to converge to a finite complex number.

Agreement on regular kernels and Theorem thm:log instead give

This is a contradiction.