The complex plane is complete, and convergence is equivalent to convergence of real and imaginary parts

Depends on

• The Euclidean metric, convergence, Cauchy sequences, and continuity on the complex plane
• For $n \ge 1$ a sequence in $\mathbb{R}^n$ converges iff each coordinate sequence converges, is Cauchy iff each coordinate sequence is Cauchy, and $\mathbb{R}^n$ is complete in every norm

Used by

• A closed subspace of ell-infinity that is not complemented Counterexample
• The integers do not have property (T) Counterexample
• Hilbert exterior powers and induced operators Definition
• Uniform convergence and the uniformly Cauchy condition for complex-valued functions, with the componentwise dictionary Definition
• At zero central parameter, distinct regular characters are inequivalent Example
• 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
• GNS representation of a continuous unitary character Example
• Pvm of a diagonal normal operator Example
• Characters of continuous functions are evaluations Lemma
• Countable compactness closes in the bidual Lemma
• Diagonal trace-class operators on ℓ²(ℕ,ℂ) 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
• Sup-norm and first-derivative bounds by the L² norm on compact subsets Lemma
• The Cauchy product of two absolutely convergent complex series converges absolutely to the product of their sums Lemma
• The disc trace space is the Hardy boundary space and the Szegő family reproduces H² 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
• A²(Ω) is closed, and the Bergman kernel is the sum over any complete orthonormal system Theorem
• Continuous functional calculus for bounded self adjoint operators Theorem
• Meromorphic continuation and intertwining identity for A(nu) Theorem
• Minimal C star unitization Theorem
• Real and complex ell one have the Schur property Theorem
• Square-Summable Complex Families Form a Hilbert Space Theorem
• The Bergman metric is positive definite on bounded domains and biholomorphically invariant 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)