PlanetPhysics/Gelfand Tornheim Theorem

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

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


 * $$\|a\| = 0$$\, if and only if\, $$a = 0$$,
 * $$\|ab\| \leqq \|a\|\cdot\|b\|$$,
 * $$\|a+b\| \leqq \|a\|+\|b\|$$,
 * $$\|ab\| = |a|\cdot\|b\|$$\, 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 $$\mathbb{C}$$ and that the valuation is the usual absolute value (the complex modulus) or some positive power of the absolute value.