The Hilbert Completion of a Real Vector Space with a Positive Semidefinite Symmetric Bilinear Form

Let $V$ be a real vector space and let $\beta:V\times V\to\mathbb{R}$ be a symmetric, bilinear, positive semidefinite map, as in Cauchy Sequences for a Positive Semidefinite Symmetric Bilinear Form: Cauchy-Schwarz, the Space of Cauchy Sequences, Convergence of Pairings, and Null Sequences. Let $C_{\beta}$ be the real vector space of $\beta$-Cauchy sequences, $\widehat\beta$ the limit pairing on $C_{\beta}$, and $[u]$ the coset of $u\in C_{\beta}$ modulo the null sequences.

1. (Completion)¶ The Hilbert completion of $(V,\beta)$ is the set $H_{\beta}=\{[u]:\ u\in C_{\beta}\}$ with the operations and pairing

which do not depend on the chosen representatives by Cauchy Sequences for a Positive Semidefinite Symmetric Bilinear Form: Cauchy-Schwarz, the Space of Cauchy Sequences, Convergence of Pairings, and Null Sequences §cosets.

2. (Canonical map)¶ The canonical map $J_{\beta}:V\to H_{\beta}$ sends $v\in V$ to the coset of the constant sequence with value $v$, which belongs to $C_{\beta}$ by Cauchy Sequences for a Positive Semidefinite Symmetric Bilinear Form: Cauchy-Schwarz, the Space of Cauchy Sequences, Convergence of Pairings, and Null Sequences §cauchy-space.