Manuscript XVIII-A@2 · XVIII-A-v2.tex · proposition · 0f4336ec

Proposition: Finite validation on a declared defect space

Exact normalization and Ward compatibility › Finite probes require coverage · lines 1174–1192 · proof included or linked

XVIII-A@2#prop:finitevalidation

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Let $D_{0}$ be a vector space of possible defects and $P : D_{0} \to C^{m}$ a linear test map. The implication

P (a) = 0 ⟹ a = 0 for every a \in D_{0}

holds exactly when $P$ is injective. No finite test map is injective on an infinite-dimensional $D_{0}$ .

Proof. The first assertion is $ker P = {0}$ . If $D_{0}$ is infinite-dimensional, choose $m + 1$ linearly independent elements. Their images are dependent, giving a nonzero element of $ker P$ .

Displayed formulas in this passage

unlabeled · line 1178
$P (a) = 0 ⟹ a = 0 for every a \in D_{0}$

Nearby in this section

section: Finite probes require coverage
example: Example: An incomplete Ward test
section: Finite probes require coverage