DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Annihilator notation and the preannihilator

Definition

Let $K = R$ or $C$ . For a normed $X$ and arbitrary subsets $M \subseteq X$ , $N \subseteq X^{*}$ , define $M^{⊥} = {f \in X^{*} : f (m) = 0 for all m \in M},^{⊥} N = {x \in X : f (x) = 0 for all f \in N} .$ Here $X^{*}$ is The dual space X^* of a normed space and its dual norm. The first notation agrees with Continuous annihilator of a linear subspace on $span M$ , since linearity makes vanishing on $M$ equivalent to vanishing on its span. The preannihilator lies in $X$ , not in $X^{* *}$ . Empty sets impose no conditions: $\emptyset^{⊥} = X^{*}$ and $^{⊥} \emptyset = X$ .

Depends on

Used by

The transpose of an injective map need not have norm-dense range Counterexample
The annihilator of a coordinate subspace Example
Annihilators and preannihilators are norm closed Lemma
Elementary kernel and range annihilator identities Lemma
Finite evaluations separate a functional from a dual subspace Lemma
Double annihilators give norm and weak-star closures Theorem
The dual of a closed subspace is a dual quotient Theorem
The dual of a quotient is its annihilator Theorem

Dependency tree · two levels

7 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Brezis, Functional Analysis, Sobolev Spaces and PDEs, §1.3, notation p.9 (standard reference, not scraped)