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.

The norm $∥ v ∥ = \sqrt{⟨ v, v ⟩}$ induced by a real or complex inner product

Definition

For a vector $v$ in a real or complex inner product space, define its inner-product norm by

$∥ v ∥ : = \sqrt{⟨ v, v ⟩} .$

Positive definiteness in Real and complex inner product spaces, with the inner product linear in the first argument makes the radicand a nonnegative real, and Existence and uniqueness of $n$ -th roots: a unique $a^{1 / n} \geq 0$ with $(a^{1 / n})^{n} = a$ supplies its unique nonnegative square root. The notation therefore defines one real number $∥ v ∥ \geq 0$ . Its norm axioms are established in The inner-product norm is definite, homogeneous, and satisfies the triangle inequality.

Depends on

Dependency tree · 2 levels

Sources