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src/fa/tvs/completion.tex

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+\section{The Hausdorff Completion}
+\label{section:tvs-complete}
+
+\begin{definition}[Hausdorff Completion of TVS]
+\label{definition:tvs-completion}
+    Let $E$ be a TVS over $K \in \RC$, then there exists $(\wh E, \iota)$ such that:
+    \begin{enumerate}
+        \item $\wh E$ is a complete Hausdorff TVS.
+        \item $\iota \in L(E; \wh E)$.
+        \item[(U)] 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:
+    \end{enumerate}
+    Moreover,
+    \begin{enumerate}
+        \item[(4)] $\iota(E)$ is dense in $\wh E$.
+    \end{enumerate}
+    The pair $(\wh E, \iota)$ is the \textbf{Hausdorff completion} of $E$.
+\end{definition}
+\begin{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 (\ref{definition:hausdorff-completion}).
+
+    Using \ref{proposition:initial-completion}, 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 \ref{proposition:hausdorff-uniform-factor}, 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
+    \[
+    \xymatrix{
+    \widehat E \times \widehat E \ar@{->}[r] & \widehat E &  & K \times \widehat E \ar@{->}[r] & \widehat E \\
+    E \times E \ar@{->}[u] \ar@{->}[r] & E \ar@{->}[u] &  & K \times E \ar@{->}[u] \ar@{->}[r] & E \ar@{->}[u]
+    }
+    \]
+
+    By continuity and the density of $\iota(E)$ in $E$, $\wh E$ with these operations forms a TVS, and $T$ is linear.
+\end{proof}
+
+\begin{remark}[{{\cite[Section 1.1]{SchaeferWolff}}}]
+	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.
+\end{remark}