False statementConstruction: AI-adaptedVerification: AI-adaptedprecheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16

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.

FALSE: On a complex inner product space, $(λ T)^{} = λ T^{}$ for every scalar

Statement

False claim. On a complex inner product space, $(λ T)^{*} = λ T^{*}$ for every scalar $λ$ and linear map $T$ .

Facts & Assumptions

Given: The identity map $I$ on standard $C$ and the scalar $λ = i$ .

[L1]

Adjoint algebra gives $(λ T)^{*} = \overline{λ} T^{*}$ , not $λ T^{*}$ (Adjoints satisfy $(S + T)^{*} = S^{*} + T^{*}$ , $(λ T)^{*} = \overline{λ} T^{*}$ , $(S T)^{*} = T^{*} S^{*}$ , and $T^{* *} = T$ ).

[L2]

The standard complex inner product is linear in its first argument and conjugate-linear in its second (The standard formulas $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ and $\sum_{k < n} x_{k} \overline{y_{k}}$ on $C^{n}$ are inner products).

Refutation

technique · counterexample

1.1L1algebra

By [L1], $(i I)^{*} = \overline{i} I^{*} = - i I$ , which is not $i I$ .

2.1step 1.1L2∎

Directly, [L2] gives $⟨ i z, w ⟩ = i ⟨ z, w ⟩ = ⟨ z, - i w ⟩$ , confirming that the adjoint scalar must be $- i$ .

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

9 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

Sheldon Axler, Linear Algebra Done Right, 4th ed., result 7.5 (standard reference, not scraped)

FALSE: On a complex inner product space, (λT)∗=λT∗(λT)∗=λT∗ for every scalar