DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-09-05

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.

A sublinear functional on a real vector space

Definition

Let $X$ be a real vector space (Vector space over a field). A function $p : X \to R$ is a sublinear functional if

$p (x + y) \leq p (x) + p (y) for all x, y \in X,$

and

$p (t x) = t p (x) for all x \in X and all t \geq 0.$

Remarks

Taking $t = 0$ shows $p (0) = 0$ .
No condition is imposed for negative scalars beyond what follows from the two displayed axioms.
A norm on a real vector space is sublinear, but a sublinear functional need not be symmetric: in general one can have $p (- x) \neq p (x)$ .

Depends on

Vector space over a field

Dependency tree · 2 levels

Sources