DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-08-28

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.

Locally bounded families of functions on a plane domain

Definition

Let $Ω \subseteq C$ be a plane domain and let $F$ be a family of functions $Ω \to C$ . The family $F$ is locally bounded when for every $a \in Ω$ there are a radius $r > 0$ and a real $M \geq 0$ such that $\overline{D} (a, r) \subseteq Ω and ∣ f (z) ∣ \leq M$ for every $f \in F$ and every $z \in \overline{D} (a, r)$ .

The bound $M$ and neighbourhood may depend on $a$ , but they are uniform over the whole family on that chosen neighbourhood.

Used by

Dependency tree · 0 levels

Nothing. This result depends on no other item in the library.

Sources

Matthias Weber, Complex Analysis, Ch. 5 §§5.1-5.2 (standard reference, not scraped)
Elias M. Stein and Rami Shakarchi, Complex Analysis, Ch. 2 §5.2 and Ch. 8 §3.2 (standard reference, not scraped)
Sheldon Axler, Paul Bourdon, and Wade Ramey, Harmonic Function Theory, Ch. 2 (standard reference, not scraped)