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

Used by

Under Dependent Choice, a bounded operator between Banach spaces is bounded below exactly when it is injective with closed range Theorem

Dependency tree · two levels

4 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Gerald Teschl, Topics in Real and Functional Analysis (standard reference, not scraped)
Haim Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations (standard reference, not scraped)