DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

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.

The graph of a linear operator with a linear domain

Definition

For normed spaces $X, Y$ , a linear subspace $D \subseteq X$ (Linear subspace of a vector space) and a linear map $A : D \to Y$ , its graph is $G (A) = {(x, A x) : x \in D} \subseteq X \times Y$ . Equip $X \times Y$ with the maximum product norm $∥ (x, y) ∥_{\max} = \max {∥ x ∥, ∥ y ∥}$ from The standard product norms on a finite product of normed spaces. The operator is closed when $G (A)$ is closed in this norm.

Depends on

Used by

A closed everywhere-defined graph need not be bounded without completeness Counterexample
A closable densely defined linear operator Definition
Differentiation on C¹[0,1] is closed and unbounded in the supremum norm Example
Closed graph theorem Theorem

Dependency tree · two levels

9 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

Buhler--Salamon, Functional Analysis, Section 2.2.2 (standard reference, not scraped)