Alphabeta MathHow 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 uniform convergence on a plane domain is the already-published compact-convergence notion
Remark
This page uses the already-published compact-convergence dictionary for local uniform convergence. Concretely, on a plane domain , a sequence of continuous maps converges locally uniformly exactly when it converges uniformly on every compact subset of (Locally uniform convergence on an open subset of the complex plane is compact convergence).
The point of the present page is not to redefine that notion, but to package it through a canonical compact exhaustion and the resulting weighted metric on function spaces.
Depends on
Used by
- Every derivative operator is continuous for locally uniform convergence on holomorphic functions Corollary
- Chordal local uniform convergence and meromorphic normality Definition
- Normal families of holomorphic functions on a plane domain Definition
- The exhaustion metric induces exactly the topology of locally uniform convergence 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
- 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)