DefinitionDefinition: AI-adaptedProof: Not applicablePipeline-generatedjudge pass (gpt-6-luna)

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.

Complex norm and complex normed space

Definition

Let $V$ be a vector space over $C$ (Vector space over a field, with the complex field of The complex numbers as $R [x] / (x^{2} + 1)$ , with the real embedding and imaginary unit $i$ ). A complex norm on $V$ is a function $N : V \to R$ such that, for all $u, v \in V$ and every $λ \in C$ :

$N (v) = 0$ if and only if $v = 0$ ;
$N (λ v) = ∣ λ ∣_{C} N (v)$ , where $∣ λ ∣_{C}$ is the complex modulus of Real and imaginary parts, complex conjugation, and modulus;
$N (u + v) \leq N (u) + N (v)$ .

A complex normed space is a pair $(V, N)$ with $V$ a complex vector space and $N$ a complex norm on $V$ . Since the codomain is $R$ , every norm value is a finite real number; extended-real values are not allowed. The homogeneity clause is for every complex scalar, not merely real scalars.

Complex norm and complex normed space

Definition

Depends on

Dependency tree · 2 levels

Sources