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

Let $V$ be a real vector space, let $\beta:V\times V\to\mathbb{R}$ be symmetric, bilinear and positive semidefinite with seminorm $\|\cdot\|_{\beta}$, as in Cauchy Sequences for a Positive Semidefinite Symmetric Bilinear Form: Cauchy-Schwarz, the Space of Cauchy Sequences, Convergence of Pairings, and Null Sequences, and let $H_{\beta}$ be the Hilbert completion of $(V,\beta)$ with pairing $\langle\cdot,\cdot\rangle_{H_{\beta}}$ and canonical map $J_{\beta}$. Real inner product spaces, their norms $|\cdot|$ and distances are those of Real Inner Product Space; real Hilbert spaces, convergence and density are as in Real Hilbert Space; continuity is continuity between metric spaces.

1. (Hilbert space)¶ With the operations and pairing of the definition, $H_{\beta}$ is a real inner product space, and it is a real Hilbert space.

2. (Isometry)¶ $J_{\beta}$ is linear, and $\langle J_{\beta}u,J_{\beta}v\rangle_{H_{\beta}}=\beta(u,v)$ for all $u,v\in V$; in particular $|J_{\beta}v|=\|v\|_{\beta}$, and $J_{\beta}v=0$ if and only if $\beta(v,v)=0$.

3. (Density)¶ For every $u=(u_{k})_{k\in\mathbb{N}}$ in the space of $\beta$-Cauchy sequences, the sequence $(J_{\beta}u_{k})_{k\in\mathbb{N}}$ converges to $[u]$ in $H_{\beta}$. In particular $J_{\beta}(V)$ is dense in $H_{\beta}$.

4. (Extension of bounded maps)¶ Let $K$ be a real Hilbert space, let $C\ge0$ be real, and let $T:V\to K$ be a linear map with $|Tv|_{K}\le C\|v\|_{\beta}$ for every $v\in V$. Then there is exactly one continuous map $\widehat{T}:H_{\beta}\to K$ with $\widehat{T}(J_{\beta}v)=Tv$ for every $v\in V$. The map $\widehat{T}$ is linear and $|\widehat{T}h|_{K}\le C|h|$ for every $h\in H_{\beta}$. If moreover $\langle Tu,Tv\rangle_{K}=\beta(u,v)$ for all $u,v\in V$, then $\langle\widehat{T}g,\widehat{T}h\rangle_{K}=\langle g,h\rangle_{H_{\beta}}$ for all $g,h\in H_{\beta}$.