DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-09-01

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 bounded operator that is bounded below

Definition

Let $T : X \to Y$ be a bounded linear operator. It is bounded below when there is a real constant $c > 0$ such that

$∥ T x ∥ \geq c ∥ x ∥ for all x \in X .$

Remarks

The inequality forces injectivity, because $∥ T x ∥ = 0$ then implies $∥ x ∥ = 0$ .
A bounded-below operator need not be surjective.

Depends on

A bounded linear operator between normed spaces

Dependency tree · 2 levels

Sources