# Gelfand–Tornheim theorem

###### Theorem.

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 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

• $\|a\|=0$  iff  $a=0$

• $\|ab\|\leqq\|a\|\cdot\|b\|$

• $\|a+b\|\leqq\|a\|+\|b\|$

• $\|ab\|=|a|\cdot\|b\|$  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 (modulus) or some positive power of the absolute value.

## References

• 1 Emil Artin: .  Lecture notes.  Mathematisches Institut, Göttingen (1959).
 Title Gelfand–Tornheim theorem Canonical name GelfandTornheimTheorem Date of creation 2013-03-22 14:11:49 Last modified on 2013-03-22 14:11:49 Owner pahio (2872) Last modified by pahio (2872) Numerical id 40 Author pahio (2872) Entry type Theorem Classification msc 12J05 Synonym Gelfand-Tornheim theorem Related topic ExtensionOfKrullValuation Related topic TopicEntryOnRealNumbers Related topic BanachAlgebra Related topic NormedAlgebra Related topic ArchimedeanOrderedFieldsAreReal Defines normed field