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

Throughout, $I=I_{h}$, $J=J_{h}$, which is $J_{\beta}$ by The Hilbert Completion of a Complex Vector Space with a Positive Semidefinite Hermitian Form §canonical-map, and $\langle\cdot,\cdot\rangle=\langle\cdot,\cdot\rangle_{H_{\beta}}$. By 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, $V$ is a real vector space by restriction of scalars, $\beta$ is symmetric, bilinear over $\mathbb{R}$ and positive semidefinite, and for all $u,v\in V$

By The Hilbert Completion is a Real Hilbert Space Containing a Dense Isometric Image, and Bounded Linear Maps Extend to It §hilbert, $H_{\beta}$ with $\langle\cdot,\cdot\rangle$ is a real Hilbert space: $\langle\cdot,\cdot\rangle$ is symmetric, bilinear and positive definite by conditions (a)-(d) of Real Inner Product Space §inner-product, and the metric $(\xi,\eta)\mapsto|\xi-\eta|$ is complete. By The Hilbert Completion is a Real Hilbert Space Containing a Dense Isometric Image, and Bounded Linear Maps Extend to It §isometry, $J$ is linear over $\mathbb{R}$ and $\langle Ju,Jv\rangle=\beta(u,v)$ for all $u,v\in V$; in particular $|Jv|=\|v\|_{\beta}$.

Preliminaries on $I$. The map $S:V\to H_{\beta}$, $Sv=J(iv)$, is linear over $\mathbb{R}$, since $i(u+v)=iu+iv$ and $i(cv)=(ic)v=(ci)v=c(iv)$ for real $c$ by conditions 7 and 5 of Vector Space over a Field and $J$ is linear over $\mathbb{R}$; and $\langle Su,Sv\rangle=\beta(iu,iv)=\beta(u,v)$ for all $u,v\in V$, by The Hilbert Completion is a Real Hilbert Space Containing a Dense Isometric Image, and Bounded Linear Maps Extend to It §isometry and the identity $\beta(iu,iv)=\beta(u,v)$ 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; in particular $|Sv|=\|v\|_{\beta}$. By The Hilbert Completion of a Complex Vector Space with a Positive Semidefinite Hermitian Form §imaginary-unit, $I$ is the only continuous map with $I(Jv)=Sv$ for every $v\in V$, so it is the map $\widehat{S}$ provided by The Hilbert Completion is a Real Hilbert Space Containing a Dense Isometric Image, and Bounded Linear Maps Extend to It §extension with $K=H_{\beta}$ and $C=1$. Hence $I$ is linear over $\mathbb{R}$, and, by the last part of that claim,

Next, $II$ and $-\mathrm{id}$ are continuous maps $H_{\beta}\to H_{\beta}$ (the first is a composite of continuous maps, and $|(-\xi)-(-\eta)|=|\xi-\eta|$). For $v\in V$, using the linearity of $J$ and the identity $i(iv)=(ii)v=(-1)v$ of condition 5 of Vector Space over a Field,

the last step by claim 5 of Elementary Identities in a Vector Space in $H_{\beta}$. The map $v\mapsto-Jv$ is linear over $\mathbb{R}$ with $|-Jv|=\|v\|_{\beta}$, and both $II$ and $-\mathrm{id}$ are continuous maps sending $Jv$ to $-Jv$; by the uniqueness part of The Hilbert Completion is a Real Hilbert Space Containing a Dense Isometric Image, and Bounded Linear Maps Extend to It §extension (with $K=H_{\beta}$, $C=1$) they coincide:

From (a), (b) and the symmetry and bilinearity of $\langle\cdot,\cdot\rangle$, for all $\xi,\eta\in H_{\beta}$

1. (Hilbert space) By The Hilbert Completion of a Complex Vector Space with a Positive Semidefinite Hermitian Form §completion, $(a+bi)\xi=a\xi+b\,I\xi$ for real $a,b$; for $b=0$ this is the real multiple $a\xi$ of $H_{\beta}$, so the complex scalar multiplication extends the real one. We check conditions 1-8 of Vector Space over a Field over $\mathbb{C}$. Conditions 1-4 involve only the addition, which is that of the real vector space $H_{\beta}$. Let $\xi,\eta\in H_{h}$ and $z,w\in\mathbb{C}$, written $z=a+bi$ and $w=c+di$ with real $a,b,c,d$ by claim 3 of Canonical Form and Arithmetic of Complex Numbers. Condition 6: $1=1+0i$, so $1\xi=\xi+0\,I\xi=\xi$ by claim 3 of Elementary Identities in a Vector Space in $H_{\beta}$. Condition 7: since $I$ is additive, $z(\xi+\eta)=a\xi+a\eta+b\,I\xi+b\,I\eta=z\xi+z\eta$. Condition 8: $z+w=(a+c)+(b+d)i$ by claim 4 of Canonical Form and Arithmetic of Complex Numbers, so $(z+w)\xi=(a+c)\xi+(b+d)I\xi=z\xi+w\xi$. Condition 5: $zw=(ac-bd)+(ad+bc)i$ by claim 4 of Canonical Form and Arithmetic of Complex Numbers, and by the $\mathbb{R}$-linearity of $I$ and (b)

So $H_{h}$ is a complex vector space, with the zero vector and the additive inverses of $H_{\beta}$.

We check conditions 1-4 of Complex Inner Product Space for $\langle\xi,\eta\rangle_{h}=\langle\xi,\eta\rangle-i\,\langle\xi,I\eta\rangle$. Conjugate symmetry: both pairings on the right are real, so by claim 1 of Properties of Complex Conjugation and Modulus and $\overline{i}=\overline{0+1i}=0-1i=-i$ (Complex Conjugate), then symmetry and (c),

Additivity in the second argument follows from the additivity of $I$ and of $\langle\xi,\cdot\rangle$. Homogeneity: for $z=a+bi$ as above, $I(z\eta)=a\,I\eta-b\,\eta$ by (b), so

Positive definiteness: by (c), $\langle\xi,\xi\rangle_{h}=\langle\xi,\xi\rangle=|\xi|^{2}$, a nonnegative real number that vanishes only for $\xi=0$, by condition (d) of Real Inner Product Space §inner-product.

Hence $H_{h}$ is a complex inner product space, and its induced norm $\lVert\xi\rVert$, the nonnegative square root of $\langle\xi,\xi\rangle_{h}=|\xi|^{2}$, equals $|\xi|$ by the uniqueness in Existence and Uniqueness of the Nonnegative Square Root. Since the additive inverses in $H_{h}$ and $H_{\beta}$ coincide, $\lVert\xi-\eta\rVert=|\xi-\eta|$: the metric of $H_{h}$ in Complex Hilbert Space is that of $H_{\beta}$, which is complete. So $H_{h}$ is a complex Hilbert space with norm $|\cdot|$. Finally $i=0+1i$, so $i\xi=0\xi+1\,I\xi=I\xi$ (by claim 3 of Elementary Identities in a Vector Space).

2. (Isometry) $J$ is additive, and for $v\in V$, $J(iv)=I(Jv)$ by The Hilbert Completion of a Complex Vector Space with a Positive Semidefinite Hermitian Form §imaginary-unit, which is $i\,Jv$ by claim 1. For $z=a+bi$ with real $a,b$ one has $zv=av+b(iv)$ by conditions 8 and 5 of Vector Space over a Field, so, by the $\mathbb{R}$-linearity of $J$,

Thus $J$ is complex-linear. For $u,v\in V$, by The Hilbert Completion is a Real Hilbert Space Containing a Dense Isometric Image, and Bounded Linear Maps Extend to It §isometry and the last identity 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,

3. (Density) $J(V)=J_{\beta}(V)$ is dense in $H_{\beta}$ by The Hilbert Completion is a Real Hilbert Space Containing a Dense Isometric Image, and Bounded Linear Maps Extend to It §dense. By claim 1, $H_{h}$ and $H_{\beta}$ are the same set with the same metric, hence the same metric topology, and density refers to that topology; so $J(V)$ is dense in $H_{h}$.

Common part of claims 4 and 5. The inner product $\langle\cdot,\cdot\rangle_{K}$ satisfies 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 by conditions 1-4 of Complex Inner Product Space; by that claim, $K$ with its real scalar multiplication and $\langle x,y\rangle_{K,\mathbb{R}}=\operatorname{Re}\langle x,y\rangle_{K}$ is a real Hilbert space $K_{\mathbb{R}}$ with norm $\lVert\cdot\rVert_{K}$, and

Let $\varepsilon=1$ in claim 4 and $\varepsilon=-1$ in claim 5. In both claims $T$ is additive and $T(cv)=c\,Tv$ for real $c$ (in claim 5 because $\overline{c}=c$ by claim 1 of Properties of Complex Conjugation and Modulus), and $T(iv)=\varepsilon i\,Tv$ (in claim 5 because $\overline{i}=-i$). So $T:V\to K_{\mathbb{R}}$ is linear over $\mathbb{R}$. As $h(v,v)$ is real, $h(v,v)=\beta(v,v)$, so $\lVert Tv\rVert_{K}^{2}\le C^{2}\beta(v,v)=(C\|v\|_{\beta})^{2}$, and since both $\lVert Tv\rVert_{K}$ and $C\|v\|_{\beta}$ are nonnegative, $\lVert Tv\rVert_{K}\le C\|v\|_{\beta}$. By The Hilbert Completion is a Real Hilbert Space Containing a Dense Isometric Image, and Bounded Linear Maps Extend to It §extension applied to $T:V\to K_{\mathbb{R}}$, there is exactly one continuous map $\widehat{T}:H_{\beta}\to K_{\mathbb{R}}$ with $\widehat{T}(Jv)=Tv$ for every $v\in V$; it is linear over $\mathbb{R}$ and $\lVert\widehat{T}\xi\rVert_{K}\le C|\xi|$. The metric of $H_{h}$ is that of $H_{\beta}$ by claim 1, and $K$ and $K_{\mathbb{R}}$ have the same norm and the same differences, hence the same metric; so the continuous maps $H_{h}\to K$ are exactly the continuous maps $H_{\beta}\to K_{\mathbb{R}}$. This gives the existence and uniqueness of $\widehat{T}$ in claims 4 and 5, its additivity, and the bound $\lVert\widehat{T}\xi\rVert_{K}\le C|\xi|$.

We show

The map $v\mapsto T(iv)$ is linear over $\mathbb{R}$, and $\lVert T(iv)\rVert_{K}\le C\|iv\|_{\beta}=C\|v\|_{\beta}$ because $\beta(iv,iv)=\beta(v,v)$ by 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. The map $\xi\mapsto\widehat{T}(I\xi)$ is continuous as a composite of continuous maps, and so is $\xi\mapsto\varepsilon i\,\widehat{T}\xi$, since $y\mapsto\varepsilon i\,y$ is additive and preserves $\lVert\cdot\rVert_{K}$ by absolute homogeneity (claim 2 of The Induced Norm is a Norm, and Induces a Metric) and $|\varepsilon i|=1$. For $v\in V$ both send $Jv$ to $T(iv)$: $\widehat{T}(I(Jv))=\widehat{T}(J(iv))=T(iv)$ by The Hilbert Completion of a Complex Vector Space with a Positive Semidefinite Hermitian Form §imaginary-unit, and $\varepsilon i\,\widehat{T}(Jv)=\varepsilon i\,Tv=T(iv)$. By the uniqueness part of The Hilbert Completion is a Real Hilbert Space Containing a Dense Isometric Image, and Bounded Linear Maps Extend to It §extension applied to $v\mapsto T(iv)$, (e) holds.

Now let $z=a+bi$ with real $a,b$. By The Hilbert Completion of a Complex Vector Space with a Positive Semidefinite Hermitian Form §completion, the $\mathbb{R}$-linearity of $\widehat{T}$, (e) and conditions 5 and 8 of Vector Space over a Field in $K$,

which is $z\,\widehat{T}\xi$ for $\varepsilon=1$ and $\overline{z}\,\widehat{T}\xi$ for $\varepsilon=-1$, since $\overline{a+bi}=a-bi$ by claim 1 of Properties of Complex Conjugation and Modulus.

Finally suppose $\langle Tu,Tv\rangle_{K}=h(u,v)$ for all $u,v$ (claim 4), respectively $\langle Tu,Tv\rangle_{K}=\overline{h(u,v)}$ (claim 5). Since $\operatorname{Re}\overline{w}=\operatorname{Re}w$ for $w\in\mathbb{C}$ (claims 1 and 2 of Properties of Complex Conjugation and Modulus), in both cases $\langle Tu,Tv\rangle_{K,\mathbb{R}}=\operatorname{Re}h(u,v)=\beta(u,v)$, and the last part of The Hilbert Completion is a Real Hilbert Space Containing a Dense Isometric Image, and Bounded Linear Maps Extend to It §extension gives $\langle\widehat{T}\xi,\widehat{T}\eta\rangle_{K,\mathbb{R}}=\langle\xi,\eta\rangle$ for all $\xi,\eta\in H_{h}$. Multiplying (e) by the real number $\varepsilon$, with $\varepsilon^{2}=1$, gives $i\,\widehat{T}\eta=\varepsilon\,\widehat{T}(I\eta)$. Hence, by (d) and the bilinearity of $\langle\cdot,\cdot\rangle_{K,\mathbb{R}}$,

4. (Extension of linear maps) Here $\varepsilon=1$. By the common part, $\widehat{T}$ exists and is unique, it is additive and complex-homogeneous, hence linear over $\mathbb{C}$, and $\lVert\widehat{T}\xi\rVert_{K}\le C|\xi|$, where $|\xi|$ is the norm of $\xi$ in $H_{h}$ by claim 1; so $C$ is a bound for $\widehat{T}$ and $\widehat{T}\in\mathcal{L}(H_{h},K)$ by Bounded Linear Maps between Complex Normed Spaces and the Operator Norm §bounded. If $\langle Tu,Tv\rangle_{K}=h(u,v)$ for all $u,v$, the last display reads $\langle\widehat{T}\xi,\widehat{T}\eta\rangle_{K}=\langle\xi,\eta\rangle-i\,\langle\xi,I\eta\rangle=\langle\xi,\eta\rangle_{h}$ by The Hilbert Completion of a Complex Vector Space with a Positive Semidefinite Hermitian Form §completion.

5. (Extension of conjugate-linear maps) Here $\varepsilon=-1$. By the common part, $\widehat{T}$ exists and is unique, it is additive, $\widehat{T}(c\xi)=\overline{c}\,\widehat{T}\xi$ and $\lVert\widehat{T}\xi\rVert_{K}\le C|\xi|$. If $\langle Tu,Tv\rangle_{K}=\overline{h(u,v)}$ for all $u,v$, the last display reads, by claim 1 of Properties of Complex Conjugation and Modulus and The Hilbert Completion of a Complex Vector Space with a Positive Semidefinite Hermitian Form §completion,