The Complex Hilbert Completion is a Complex Hilbert Space Containing a Dense Isometric Image, and Bounded Complex-Linear Maps Extend to It

In the setting of Complex Hilbert Spaces and Bounded Linear Maps: Standing Notation, 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, and let $H_{h}$, $\langle\cdot,\cdot\rangle_{h}$, $I_{h}$ and $J_{h}$ be the complex Hilbert completion of $(V,h)$, its pairing, the extended multiplication by $i$ and the canonical map; $|\cdot|$ is the norm of the underlying real Hilbert completion $H_{\beta}$, $\beta=\operatorname{Re}h$.

1. (Hilbert space)¶ $H_{h}$, with its addition, its multiplication by complex scalars and $\langle\cdot,\cdot\rangle_{h}$, is a complex Hilbert space, whose induced norm is $|\cdot|$; and $I_{h}\xi=i\xi$ for every $\xi\in H_{h}$.

2. (Isometry)¶ $J_{h}$ is complex-linear, and $\langle J_{h}u,J_{h}v\rangle_{h}=h(u,v)$ for all $u,v\in V$.

3. (Density)¶ $J_{h}(V)$ is dense in $H_{h}$.

In claims 4 and 5, $K$ is a complex Hilbert space, $C\ge0$ is real, and $T:V\to K$ is additive with $\lVert Tv\rVert_{K}^{2}\le C^{2}h(v,v)$ for every $v\in V$.

4. (Extension of linear maps)¶ If $T$ is complex-linear, there is exactly one continuous map $\widehat{T}:H_{h}\to K$ with $\widehat{T}(J_{h}v)=Tv$ for every $v\in V$; it belongs to $\mathcal{L}(H_{h},K)$, with bound $C$; and if moreover $\langle Tu,Tv\rangle_{K}=h(u,v)$ for all $u,v\in V$, then $\langle\widehat{T}\xi,\widehat{T}\eta\rangle_{K}=\langle\xi,\eta\rangle_{h}$ for all $\xi,\eta\in H_{h}$.

5. (Extension of conjugate-linear maps)¶ If $T(cv)=\overline{c}\,Tv$ for all $c\in\mathbb{C}$ and $v\in V$, there is exactly one continuous map $\widehat{T}:H_{h}\to K$ with $\widehat{T}(J_{h}v)=Tv$ for every $v\in V$; it is additive, satisfies $\widehat{T}(c\xi)=\overline{c}\,\widehat{T}\xi$ and $\lVert\widehat{T}\xi\rVert_{K}\le C\,|\xi|$ for all $c\in\mathbb{C}$ and $\xi\in H_{h}$; and if moreover $\langle Tu,Tv\rangle_{K}=\overline{h(u,v)}$ for all $u,v\in V$, then $\langle\widehat{T}\xi,\widehat{T}\eta\rangle_{K}=\overline{\langle\xi,\eta\rangle_{h}}$ for all $\xi,\eta\in H_{h}$.