Proposition: Finite-probe coverage

Let $Z\subseteq Y$ be a proved finite-dimensional residual subspace, and let $P:Y\to {\mathbb{R}}^k$ be a declared linear probe map. Exact vanishing of all probes forces $r=0$ for every $r\in Z$ if and only if $P{|}_Z$ is injective.

When it is injective, there is a finite constant $C_Z$ such that

Thus, if $r(D)\in Z$, $\Vert P\hat{r}{\Vert }_{\infty }\le u$, and (eq:residual-error) holds, then

Proof. Injectivity is precisely the assertion that the only element of $Z$ in the probe kernel is zero. If $Z\ne \{0\}$, the continuous function $r\mapsto \Vert Pr{\Vert }_{\infty }$ has a positive minimum on the compact unit sphere of $Z$. Its reciprocal gives $C_Z$. For $Z=\{0\}$, the assertion is immediate. Finally,