8.7 The Hausdorff Completion
Definition 8.7.1 (Hausdorff Completion of TVS). Let $E$ be a TVS over $K \in \RC$, then there exists $(\wh E, \iota)$ such that:
$\wh E$ is a complete Hausdorff TVS.
$\iota \in L(E; \wh E)$.
For any $(F, T)$ satisfying (1) and (2), there exists a unique $\ol{T}\in L(\wh E; F)$ such that the following diagram commutes:
Moreover,
$\iota(E)$ is dense in $\wh E$.
The pair $(\wh E, \iota)$ is the Hausdorff completion of $E$.
Proof. All claims of (1), (2), (U), and (4), except the linearity of maps and the fact that $\wh E$ is a TVS is proven via the Hausdorff completion.
Using Proposition 5.6.5, identify $\wh E \times \wh E$ with $\wh{E \times E}$ and $K \times \wh E$ with $\wh{K \times E}$ as uniform spaces. By Proposition 5.6.4, there exists operations $\wh E \times \wh E \to \wh E$ and $K \times \wh E \to \wh E$ such that the following diagrams commute
By continuity and the density of $\iota(E)$ in $E$, $\wh E$ with these operations forms a TVS, and $T$ is linear.$\square$
Remark 8.7.2. The Hausdorff completion works in general with arbitrary valuated fields. Though the completion yields a TVS over the completion of the field, the field need not to be complete.