Coordinate Isometry Determined by a Finite Orthonormal Basis

Let $V$ together with $\langle\cdot,\cdot\rangle$ be a \reftext{def:complex-inner-product-space-2026a}{complex inner product space} with \reftext{lem:vector-space-basic-identities-2026a}{zero vector} $0_{V}$ and \reftext{def:inner-product-norm-2026a}{induced norm} $\lVert\cdot\rVert$, and let $d$ be the map with $d(u,v)=\lVert u-v\rVert$, which is a \reftext{def:metric-space-2026a}{metric} on $V$ by claim 3 of \ref{lem:inner-product-norm-is-norm-2026a}. Here $u-v$ abbreviates $u+(-v)$, with the additive inverse of \ref{lem:vector-space-basic-identities-2026a}.

Let $n$ be a \reftext{def:natural-numbers-2026a}{natural number}, let $[p]$ denote the \reftext{def:initial-segment-natural-numbers-2026a}{initial segment} determined by a natural number $p$, and let $e\in V^{n}$ be an \reftext{def:finite-tuple-power-2026a}{$n$-tuple} in $V$ that is an \reftext{def:orthonormal-basis-2026b}{orthonormal basis} of $V$, with components $e_{k}$. Write $2n=n+n$, let $\mathbb{R}^{2n}$ be the \reftext{def:euclidean-space-rn-2026a}{Euclidean space} of that dimension, and let $d_{E}$ be the \reftext{def:euclidean-distance-rn-2026a}{Euclidean distance} on it, which is a metric by \ref{thm:euclidean-distance-is-metric-rn-2026a}. For a \reftext{def:complex-numbers-2026a}{complex number} $z$, let $\operatorname{Re}z$ and $\operatorname{Im}z$ be its \reftext{def:complex-real-imaginary-part-2026a}{real and imaginary parts}.

Every $i\in[2n]$ either lies in $[n]$ or is of the form $i=n+k$ for exactly one $k\in[n]$, by claims 6 and 7 of \ref{lem:order-natural-numbers-2026a}. Let $\Phi:V\to\mathbb{R}^{2n}$ be the map sending $u\in V$ to the point of $\mathbb{R}^{2n}$ whose coordinates are

Then the following hold.

\textbf{1. (Bijection)} $\Phi$ is a \reftext{def:bijection-sets-2026a}{bijection} from $V$ onto $\mathbb{R}^{2n}$.

\textbf{2. (Isometry)} $d_{E}(\Phi(u),\Phi(v))=d(u,v)$ for all $u,v\in V$.

\textbf{3. (The unit sphere is compact)} Equip $V$ with the collection of all subsets that are \reftext{def:open-subset-metric-space-2026a}{open in $(V,d)$}, a topology by \ref{thm:metric-open-sets-form-topology-2026a}. Then the set

of \reftext{def:unit-vector-2026a}{unit vectors} of $V$ is nonempty and \reftext{def:compact-space-and-subset-2026a}{compact in $V$}.