DefinitionDefinition: AI-adaptedProof: Not applicablePipeline-generated

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.

Equivalent complex norms

Definition

Let $M$ and $N$ be complex norms (Complex norm and complex normed space) on the same complex vector space $V$ . They are equivalent complex norms if there are positive real constants $c, C \in R$ such that

c M (v) \leq N (v) \leq C M (v) for every v \in V .

Both inequalities and the positivity of $c, C$ are in the real complete ordered field (Complete ordered field (least-upper-bound property), with completeness established in The Cauchy-sequence reals have the least-upper-bound property); the vector scalar field remains $C$ . No order comparison of complex numbers is intended.

Equivalent complex norms

Definition

Depends on

Dependency tree · 2 levels

Sources