Updated Schaefer & Wolff citations.
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By continuity and the density of $\iota(E)$ in $E$, $\wh E$ with these operations forms a TVS, and $T$ is linear.
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\end{proof}
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\begin{remark}[{{\cite[Section 1.1]{SchaeferWolff}}}]
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\begin{remark}[{{\cite[Section I.1]{SchaeferWolff}}}]
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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.
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\end{remark}
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