Coordinate Isometry Determined by a Finite Orthonormal Basis

Let $V$ together with $\langle\cdot,\cdot\rangle$ be a complex inner product space with zero vector $0_{V}$ and induced norm $\lVert\cdot\rVert$, and let $d$ be the map with $d(u,v)=\lVert u-v\rVert$, which is a metric on $V$ by claim 3 of The Induced Norm is a Norm, and Induces a Metric. Here $u-v$ abbreviates $u+(-v)$, with the additive inverse of Elementary Identities in a Vector Space.

Let $n$ be a natural number, let $[p]$ denote the initial segment determined by a natural number $p$, and let $e\in V^{n}$ be an $n$-tuple in $V$ that is an orthonormal basis of $V$, with components $e_{k}$. Write $2n=n+n$, let $\mathbb{R}^{2n}$ be the Euclidean space of that dimension, and let $d_{E}$ be the Euclidean distance on it, which is a metric by $\mathbb{R}^n$ with the Euclidean Distance Is a Complete Metric Space with Coordinatewise Convergence. For a complex number $z$, let $\operatorname{Re}z$ and $\operatorname{Im}z$ be its 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 Properties of the Order on the Natural Numbers. Let $\Phi:V\to\mathbb{R}^{2n}$ be the map sending $u\in V$ to the point of $\mathbb{R}^{2n}$ whose coordinates are

$$\Phi(u)_{k}=\operatorname{Re}\langle e_{k},u\rangle,\qquad \Phi(u)_{n+k}=\operatorname{Im}\langle e_{k},u\rangle\qquad\text{for }k\in[n].$$

Then the following hold.

1. (Bijection) $\Phi$ is a bijection from $V$ onto $\mathbb{R}^{2n}$.

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

3. (The unit sphere is compact) Let $\mathcal{T}_{d}$ be the collection of all subsets of $V$ that are open in $(V,d)$, which is a topology on $V$ by The Open Subsets of a Metric Space Form a Topology. Then the set

$$S=\{u\in V:\lVert u\rVert=1\}$$

of unit vectors of $V$ is nonempty and compact in $(V,\mathcal{T}_{d})$.