False statementConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-29

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FALSE: every complex vector space has a preferred real form

Statement

Every nonzero complex vector space carries a real form singled out by the complex structure alone, in the precise sense that the real form is invariant under every complex-linear automorphism.

Facts & Assumptions

Given: A nonzero complex vector space $W$ and a real form $W_{0} \subseteq W$ .

[L1]

A real form is the fixed space of a conjugation, and its complexification recovers $W$ ; in particular every $w \in W$ has a unique expression $w = u + i v$ with $u, v \in W_{0}$ (Real forms of a complex vector space correspond exactly to conjugations, The fixed points of a conjugation form a real vector space whose complexification recovers the ambient complex space).

Refutation

technique · direct

1.1

Multiplication by $i$ is a complex-linear automorphism of $W$ . If it preserved $W_{0}$ , then $i u \in W_{0}$ for every $u \in W_{0}$ .

givenalgebra

2.1

Choose $0 \neq u \in W_{0}$ . Under the preservation assumption of step 1.1, $i u \in W_{0}$ , so $i u = (i u) + i 0 = 0 + i u$ would be two decompositions of the same vector with real and imaginary parts in $W_{0}$ , contradicting uniqueness in [L1].

L1step 1.1contradiction

3.1

Thus no real form of a nonzero complex vector space is invariant under all complex-linear automorphisms. The complex structure alone therefore singles out no preferred real form, and the claim is false.

step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

8 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)