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Holomorphic functions form a closed subspace for locally uniform convergence
Statement
The holomorphic functions on a plane domain form a closed subspace of for the topology of locally uniform convergence. Equivalently, a -limit of holomorphic functions is holomorphic.
Facts & Assumptions
Given: A sequence of holomorphic functions converging in the exhaustion metric to a continuous limit .
Convergence in the exhaustion metric is exactly local uniform convergence (The exhaustion metric induces exactly the topology of locally uniform convergence).
A locally uniform limit of holomorphic functions is holomorphic (Locally uniform limits of holomorphic functions are holomorphic and their derivatives converge locally uniformly).
Proof
Fact [L1] turns the metric convergence into local uniform convergence on the domain.
Applying [L2] shows that the limit function is holomorphic. Hence holomorphic functions form a closed subspace for local uniform convergence.
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Used by
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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)