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The complex plane is complete, and convergence is equivalent to convergence of real and imaginary parts
Statement
The metric space is complete. A sequence converges to exactly when and . The conventions and prerequisite facts used below are recorded in The Euclidean metric, convergence, Cauchy sequences, and continuity on the complex plane, For a sequence in converges iff each coordinate sequence converges, is Cauchy iff each coordinate sequence is Cauchy, and is complete in every norm.
Facts & Assumptions
Given: A complex sequence .
Proof
The complex metric is the Euclidean metric on .
Apply componentwise convergence and completeness in the published Euclidean-space theorem.
Depends on
Used by
- A closed subspace of ell-infinity that is not complemented Counterexample
- Uniform convergence and the uniformly Cauchy condition for complex-valued functions, with the componentwise dictionary Definition
- Diagonal Schatten class criteria on ell two Example
- Fourier transform as the Gelfand transform of an LCA group algebra Example
- Gelfand transform of ell one of Z Example
- Pvm of a diagonal normal operator Example
- Characters of continuous functions are evaluations Lemma
- Countable compactness closes in the bidual Lemma
- Eberlein–Šmulian metrization on the relevant dual ball Lemma
- Goursat bisection selects nested triangles retaining one quarter of the boundary-integral magnitude, with halving diameters and a one-point intersection Lemma
- Real and complex c₀ are Banach Lemma
- The Cauchy product of two absolutely convergent complex series converges absolutely to the product of their sums Lemma
- The finitely additive integral is well-defined and isometric Lemma
- A sequence of complex-valued functions converges uniformly if and only if it is uniformly Cauchy Theorem
- Continuous functional calculus for bounded self adjoint operators Theorem
- Minimal C star unitization Theorem
- Real and complex ell one have the Schur property Theorem
Dependency tree · two levels
28 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
- J. Lebl, Basic Analysis I: Complex Numbers and the Complex Exponential (standard reference, not scraped)