The Euclidean metric, convergence, Cauchy sequences, and continuity on the complex plane

Depends on

• Real and imaginary parts, complex conjugation, and modulus
• The $p$-norms $\lVert x\rVert_p$ for rational $p \ge 1$, and $\lVert x\rVert_\infty$
• Each $\lVert\cdot\rVert_p$ is a norm on $\mathbb{R}^n$, and the induced metrics are exactly $d_1$, $d_2$ and $d_\infty$ of the published metric-spaces page
• Metric space: $d(x,y) = 0$ iff $x = y$, symmetry, and the triangle inequality; pseudometric and ultrametric
• Convergence of a sequence in a metric space: $x_k \to x$ iff $d(x_k, x) \to 0$ in $\mathbb{R}$
• Cauchy sequence in a metric space
• Continuity of a map between metric spaces, at a point and globally, in the $\varepsilon$-$\delta$ form

Used by

• Complex differentiability at a point implies continuity there Corollary
• Every nonconstant entire function has dense image in the complex plane Corollary
• Spectrum of a compact operator is countable with only zero as possible accumulation Corollary
• f(x+iy)=eˣ(cos 2y+i sin 2y) is continuous, satisfies f(z+w)=f(z)f(w) and f(1)=e, but is not the standard complex exponential Counterexample
• The disc algebra is unital and separating but not self-adjoint or dense Counterexample
• Complex differentiability at a point, the complex derivative, holomorphic functions, and entire functions Definition
• Complex series, absolute convergence, complex power series, and radius of convergence Definition
• Self-adjoint complex function algebras, unitality, and point separation Definition
• Spectrum and resolvent of a bounded operator Definition
• Square-summable families on an arbitrary index set and the space ℓ²(I) Definition
• Topological vector spaces over the real and complex fields Definition
• Uniform convergence and the uniformly Cauchy condition for complex-valued functions, with the componentwise dictionary Definition
• Trigonometric polynomials are uniformly dense on the unit circle Example
• A root-free complex polynomial gives nullhomotopic normalized circle loops Lemma
• Character space of generated normal algebra is operator spectrum Lemma
• The real-valued part of a point-separating self-adjoint complex function algebra is separating and has the same common zeros Lemma
• ℂ=ℝ[x]/(x²+1) as the Euclidean plane and as a normed real algebra: what the identification preserves Remark
• Complex Stone–Weierstrass dichotomy for separating self-adjoint algebras; the unital case is dense Theorem
• Fundamental theorem of algebra by Liouville's theorem Theorem
• Riesz schauder spectrum of a compact operator Theorem
• Schauder compact adjoint theorem Theorem
• Spectral theorem for compact self adjoint operators Theorem
• The complex plane is complete, and convergence is equivalent to convergence of real and imaginary parts Theorem
• There is no continuous logarithm on all of ℂ∖{0} Theorem

Dependency tree · two levels

46 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)