The Hilbert Completion of a Complex Vector Space with a Positive Semidefinite Hermitian Form

Let $\mathbb{C}\supseteq\mathbb{R}$ be the field of complex numbers, with imaginary unit $i$, and let $V$ be a complex vector space. Let $h:V\times V\to\mathbb{C}$ satisfy the hypotheses of Bounded Linear Maps between Complex Inner Product Spaces: the Least Bound, Operations, the Underlying Real Structure, Adjoints, Completeness and the Quadratic-Form Bound §real-structure: $h$ is additive and complex-homogeneous in its second argument, $h(v,u)=\overline{h(u,v)}$, and $h(v,v)$ is a real number with $0\le h(v,v)$, for all $u,v\in V$. By that claim, $V$ is a vector space over $\mathbb{R}$ by restriction of scalars, and $\beta=\operatorname{Re}h$ is symmetric, bilinear over $\mathbb{R}$ and positive semidefinite, with $\beta(iv,iv)=\beta(v,v)$ for every $v\in V$. Let $H_{\beta}$ be the Hilbert completion of $(V,\beta)$, with pairing $\langle\cdot,\cdot\rangle_{H_{\beta}}$, norm $|\cdot|$ and canonical map $J_{\beta}$; it is a real Hilbert space by The Hilbert Completion is a Real Hilbert Space Containing a Dense Isometric Image, and Bounded Linear Maps Extend to It §hilbert, and $\|\cdot\|_{\beta}$ is the seminorm of Cauchy Sequences for a Positive Semidefinite Symmetric Bilinear Form: Cauchy-Schwarz, the Space of Cauchy Sequences, Convergence of Pairings, and Null Sequences.

1. (Multiplication by $i$)¶ The map $v\mapsto J_{\beta}(iv)$ from $V$ to $H_{\beta}$ is linear over $\mathbb{R}$, since $J_{\beta}$ is linear by The Hilbert Completion is a Real Hilbert Space Containing a Dense Isometric Image, and Bounded Linear Maps Extend to It §isometry, and for every $v\in V$ one has $|J_{\beta}(iv)|=\|iv\|_{\beta}$ by the same claim, which equals $\|v\|_{\beta}$ since $\beta(iv,iv)=\beta(v,v)$. Hence, by The Hilbert Completion is a Real Hilbert Space Containing a Dense Isometric Image, and Bounded Linear Maps Extend to It §extension with $C=1$, there is exactly one continuous map $I_{h}:H_{\beta}\to H_{\beta}$ with $I_{h}(J_{\beta}v)=J_{\beta}(iv)$ for every $v\in V$.

2. (Completion)¶ The complex Hilbert completion of $(V,h)$ is the set $H_{h}=H_{\beta}$ with the addition and the multiplication by real scalars of $H_{\beta}$, with the multiplication by complex scalars

which is well defined because every complex number is $a+bi$ for exactly one pair of real numbers $a,b$ by Canonical Form and Arithmetic of Complex Numbers, and with the pairing

3. (Canonical map)¶ The canonical map of the complex Hilbert completion is $J_{h}=J_{\beta}:V\to H_{h}$.