TheoremStatement: Literature-sourcedProof: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-28

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Continuous complex-valued functions on a plane domain are complete for an exhaustion metric

Statement

For a plane domain $Ω$ , the space $C (Ω, C)$ of continuous complex-valued functions is complete for the canonical exhaustion metric.

Facts & Assumptions

Given: A plane domain $Ω$ , its canonical exhaustion $(K_{n})$ , and a $d_{K}$ -Cauchy sequence $(f_{m})$ in $C (Ω, C)$ .

[L1]

A sequence of complex-valued functions is uniformly Cauchy on a set if and only if it converges uniformly on that set (A sequence of complex-valued functions converges uniformly if and only if it is uniformly Cauchy).

[L2]

A uniform limit of continuous complex-valued functions is continuous (A uniform limit of continuous complex-valued functions is continuous).

[L3]

The canonical exhaustion is nested and its interiors exhaust $Ω$ (Every plane domain has the canonical nested compact exhaustion by distance and radius cutoffs).

Proof

technique · direct

1.1

On each $K_{n}$ , the $d_{K}$ -Cauchy property makes $(f_{m} ∣_{K_{n}})$ uniformly Cauchy; if $K_{n} = \emptyset$ take the zero function there, and otherwise [L1] gives a uniform limit $g_{n} : K_{n} \to C$ , which is continuous by [L2].

L1L2given

1.2

The nesting of [L3] makes these limits compatible on overlaps, so they glue to a single function $f : Ω \to C$ . Because every point of $Ω$ lies in the interior of some $K_{n}$ by [L3], the glued function is continuous there.

L3given

2.1

Uniform convergence $f_{m} ∣_{K_{n}} \to f ∣_{K_{n}}$ on each $K_{n}$ , together with the usual finite-head and geometric-tail estimate in the defining series, gives $d_{K} (f_{m}, f) \to 0$ . Thus $C (Ω, C)$ is complete for the exhaustion metric.

givenalgebra∎

Depends on

Used by

Holomorphic functions form a closed subspace for locally uniform convergence Corollary

Dependency tree · two levels

11 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)