Complex $m$-space and its real coordinate dictionary

Fix a natural number $m\ge 1$. Complex $m$-space ${\mathbb{C}}^m$ is the set of functions $m\to \mathbb{C}$, so a point $z$ has coordinates $z_k$ for $k<m$, indexed from $0$ exactly as ${\mathbb{R}}^n$ is in this library. With coordinatewise addition and multiplication by complex scalars it is a vector space over the field $\mathbb{C}$ (Vector space over a field, $\mathbb{C}=\mathbb{R}[x]\mathrm{/}(x^2+1)$ is a field, every element is uniquely $a+bi$, and every nonzero element has inverse $(a-bi)\mathrm{/}(a^2+b^2)$), with the standard basis $e_0,\dots ,e_{m-1}$ of The standard list $e:n\to F^n$ with $e_i(i)=1_F$ and $e_i(j)=0_F$ for $j\mathrm{\ne }i$ is an ordered basis of $F^n$; hence ${\dim }_F{F}^n=n$, and $F^0$ is the zero space with basis $\mathrm{\varnothing }$ and dimension $0$.

The coordinate identification. Writing $z_k=x_k+i{y}_k$ with $x_k,y_k$ real (Real and imaginary parts, complex conjugation, and modulus), define

$\mathrm{\Phi }:{\mathbb{C}}^m\to {\mathbb{R}}^{2m},\mathrm{\Phi }(z)=(x_0,y_0,x_1,y_1,\dots ,x_{m-1},y_{m-1})\mathrm{.}$

The interleaved ordering is the one used throughout this page; the ordering that groups all real parts before all imaginary parts is a different bijection, and nothing below is stated for it. $\mathrm{\Phi }$ is a bijection and is $\mathbb{R}$-linear.

Norms agree. Put $\parallel z\parallel :=({\sum }_{k<m}\mathrm{\mid }{z}_k{\mathrm{\mid }}^2{)}^{1\mathrm{/}2}$. Since $\mathrm{\mid }{z}_k{\mathrm{\mid }}^2=x_k^2+y_k^2$, this is the Euclidean norm $\parallel \mathrm{\Phi }(z){\parallel }_2$ of The $p$-norms $\parallel x{\parallel }_p$ for rational $p\ge 1$, and $\parallel x{\parallel }_{\mathrm{\infty }}$ and The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$, and it is a norm on the real vector space underlying ${\mathbb{C}}^m$ in the sense of A norm on a real vector space, the induced metric, and the dictionary with the metric axioms. Consequently $\parallel z-w\parallel =\parallel \mathrm{\Phi }(z)-\mathrm{\Phi }(w){\parallel }_2$, so the metric of ${\mathbb{C}}^m$, its balls (Open ball, closed ball and sphere in a metric space), its open sets, its convergent sequences, its Cauchy sequences and its continuous maps are verbatim those of ${\mathbb{R}}^{2m}$ under $\mathrm{\Phi }$. In particular convergence and continuity are coordinatewise (A vector-valued function has a limit, or is continuous, if and only if each of its components does; with the algebra of continuous vector-valued functions, Vector-valued functions $f:A\to {\mathbb{R}}^m$, their limits and continuity, with the dictionary to the metric notions), ${\mathbb{C}}^m$ is complete (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), and a subset of ${\mathbb{C}}^m$ is compact exactly when it is closed and bounded (Heine-Borel in ${\mathbb{R}}^n$: with the Euclidean metric a subset of ${\mathbb{R}}^n$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line).

At $m=1$ this is the published plane dictionary. For $m=1$ the map $\mathrm{\Phi }$ is the bijection of $\mathbb{C}=\mathbb{R}[x]\mathrm{/}(x^2+1)$ as the Euclidean plane and as a normed real algebra: what the identification preserves and every clause above reduces to a clause recorded there. Openness, connectedness and real total differentiability on ${\mathbb{C}}^m$ are always read through $\mathrm{\Phi }$, exactly as that remark reads them through its own identification.

What $\mathrm{\Phi }$ does not carry. $\mathrm{\Phi }$ respects the additive and the real scalar structure but not multiplication by $i$ in any way visible to a general $\mathbb{R}$-linear map of ${\mathbb{R}}^{2m}$: an $\mathbb{R}$-linear map of ${\mathbb{C}}^m$ need not be $\mathbb{C}$-linear. That distinction is the whole content of the criterion the page proves next, and it is why "linear" is always qualified below.