DefinitionDefinition: Literature-sourcedProof: Not applicablejudge 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.

Real and complex inner product spaces, with the inner product linear in the first argument

Definition

Let $F$ be either $R$ or $C$ , with conjugation equal to the identity on $R$ and with complex conjugation as in Real and imaginary parts, complex conjugation, and modulus. An inner product on an $F$ -vector space $V$ is a function $⟨ \cdot, \cdot ⟩ : V \times V \to F$ such that for all $u, v, w \in V$ and $a, b \in F$ :

$⟨ a u + b v, w ⟩ = a ⟨ u, w ⟩ + b ⟨ v, w ⟩$ ;
$⟨ u, v ⟩ = \overline{⟨ v, u ⟩}$ ;
$⟨ v, v ⟩$ is real and nonnegative, and $⟨ v, v ⟩ = 0$ if and only if $v = 0$ .

The first two clauses imply conjugate-linearity in the second argument: $⟨ u, a v + b w ⟩ = \overline{a} ⟨ u, v ⟩ + \overline{b} ⟨ u, w ⟩$ . This is the linear-first convention of Sesquilinear and Hermitian forms over a field with an involution, using the convention linear in the first variable. A vector space equipped with an inner product is an inner product space.

Real and complex inner product spaces, with the inner product linear in the first argument

Definition

Depends on

Dependency tree · 2 levels

Sources