No Non-Trivial Norm on Rational Numbers is Complete

Theorem

No non-trivial norm on the set of the rational numbers is complete.

Proof

By P-adic Norm not Complete on Rational Numbers, no $p$-adic norm $\norm{\,\cdot\,}_p$ on the set of the rational numbers, for any prime $p$, is complete.

By Rational Number Space is not Complete Metric Space, the absolute value $\size{\,\cdot\,}$ on the set of the rational numbers is not complete.

By Norm is Complete Iff Equivalent Norm is Complete, no norm is complete if it is equivalent to either the absolute value $\size{\,\cdot\,}$ or the $p$-adic norm $\norm{\,\cdot\,}_p$ for some prime $p$.

By Ostrowski's Theorem, every non-trivial norm is equivalent to either the absolute value $\size{\,\cdot\,}$ or the $p$-adic norm $\norm{\,\cdot\,}_p$ for some prime $p$.

The result follows.

$\blacksquare$

Sources

• 1997: Fernando Q. Gouvea: p-adic Numbers: An Introduction ... (previous) ... (next): $\S 3.2$ Completions, Lemma $3.2.3$