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is the real coordinate plane, with coordinate arithmetic
Statement
The map is a bijection. Under it, and
Facts & Assumptions
Given: The complex field in its quotient construction.
Every complex number has a unique form , and addition and multiplication have the displayed coordinate formulas ( is a field, every element is uniquely , and every nonzero element has inverse ).
Proof
Uniqueness in [F1] makes well-defined and injective.
For every , the complex number maps to , so is surjective.
Applying to the addition formula in [F1] gives the first coordinate identity.
Applying to the multiplication formula in [F1] gives the second coordinate identity.
Depends on
Used by
- A finite-dimensional normed subspace is closed Corollary
- Every finite-dimensional normed space is Banach Corollary
- Holomorphic functions are real analytic and smooth in their two real coordinates Corollary
- The edgewise Riemann integral around a complex triangle for an integrable pullback Definition
- FALSE: real differentiability as a map ℝ²→ℝ² implies complex differentiability; conjugation is the counterexample False statement
- Plane similarities are complex or conjugate-complex multiplications; the orientation-preserving ones are exactly the nonzero complex multiplications Lemma
- The real-valued part of a point-separating self-adjoint complex function algebra is separating and has the same common zeros Lemma
- Vector addition and scalar multiplication are continuous in a normed space Lemma
- ℂ=ℝ[x]/(x²+1) as the Euclidean plane and as a normed real algebra: what the identification preserves Remark
- A Banach space has no countably infinite Hamel basis Theorem
- A chosen algebraic basis identifies a finite-dimensional normed space with a coordinate space Theorem
- A jointly continuous finite-interval parameter integral of holomorphic functions is holomorphic Theorem
- A normed space is locally compact if and only if it is finite-dimensional Theorem
- All norms on a finite-dimensional complex normed space are equivalent Theorem
- Complex differentiability is equivalent to real total differentiability together with a complex-linear derivative, with ∂_z̄f=0, or with the Cauchy–Riemann equations Theorem
- The closed unit ball is compact if and only if the normed space is finite-dimensional Theorem
Dependency tree · two levels
6 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
- R. K. Srivastava, Complex Analysis lecture notes (standard reference, not scraped)