DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedaudited 2026-08-29

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.

The fixed real form of a conjugation

Definition

Let $σ$ be a conjugation on the complex vector space $W$ (Conjugations and real structures on a complex vector space). Its fixed real form is the real subspace

$W^{σ} : = {w \in W : σ (w) = w} \subseteq W_{R},$

where $W_{R}$ is the realification of $W$ . That $W^{σ}$ is a real subspace is immediate: it contains $0$ , is closed under addition because $σ$ is additive, and is closed under real scalars because $σ (r w) = \overline{r} σ (w) = r w$ for $r \in R$ .

Remarks

The same complex vector space can carry different conjugations with different fixed real forms; a fixed real form is extra structure attached to the choice of $σ$ , not canonical data of $W$ alone.

Depends on

Conjugations and real structures on a complex vector space

Used by

Dependency tree · two levels

4 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

Mikhail Troshkin, Real-complex linear algebra and abelian varieties (standard reference, not scraped)