Theorem: Normalized continuous separator

For every nonzero real $f$, there is a continuous linear functional ${\mathcal{Q}}_f:{\mathcal{P}}_{\Sigma }\to \mathbb{C}$ such that

In particular,

although their coefficient kernels converge strongly.

If two supplied numerical evaluations of ${\mathcal{Q}}_f$ have absolute errors at most ${ϵ}_0,{ϵ}_1$ on a bulk element and on $B_f$, their reported separation is at least $1-{ϵ}_0-{ϵ}_1$.

Proof. Choose $\chi \in {C_c}^{\infty }(\Sigma )$ equal to one on the trace support in Lemma lem:positive, and set

This is a continuous coordinate of the graph topology. Equations (eq:Btrace) and (eq:positive) give its value one on $B_f$; Proposition prop:bulkinclusion gives zero on the bulk. No net of bulk elements can converge to $B_f$ in this topology.

Strong convergence of the mollified compact kernel follows by transposition: convolution of a bounded set of smooth tests converges uniformly with all finitely required derivatives on the common compact set. The error assertion is the scalar triangle inequality.