PlanetPhysics/Gelfand Tornheim Theorem

\htmladdnormallink{theorem {http://planetphysics.us/encyclopedia/Formula.html}.}\, Any normed [[../CosmologicalConstant2/|field]] is isomorphic either to the field $ℝ$ of real numbers or to the field $ℂ$ of complex numbers.\\

The normed field means here a field $K$ having a subfield $R$ isomorphic to $ℝ$ and satisfying the following: \, There is a mapping $\Vert \cdot \Vert$ from $K$ to the set of non-negative reals such that

• $\Vert a\Vert =0$\, if and only if\, $a=0$,
• $\Vert ab\Vert \leqq \Vert a\Vert \cdot \Vert b\Vert$,
• $\Vert a+b\Vert \leqq \Vert a\Vert +\Vert b\Vert$,
• $\Vert ab\Vert =|a|\cdot \Vert b\Vert$\, when\, $a\in R$\, and\, $b\in K$.

Using the Gelfand--Tornheim theorem, it can be shown that the only fields with archimedean valuation are isomorphic to subfields of $ℂ$ and that the valuation is the usual absolute value (the complex modulus) or some positive power of the absolute value.