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'Gelfand-Tornheim theorem'
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| Title of object: |
Gelfand-Tornheim theorem |
| Canonical Name: |
GelfandTornheimTheorem |
| Type: |
Theorem |
| Created on: |
2004-02-26 04:37:19 |
| Modified on: |
2004-10-19 04:33:35 |
| Classification: |
msc:12J05 |
| Keywords: |
real numbers, complex numbers |
| Defines: |
normed field |
Preamble:
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Content:
\textbf{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 {\em 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
\begin{itemize}
\item $\|a\| = 0$ \,iff \,$a = 0$
\item $\|ab\| \leqq \|a\|\cdot\|b\|$
\item $\|a+b\| \leqq \|a\|+\|b\|$
\item $\|ab\| = |a|\cdot\|b\|$ \,when \,$a \in R$ \,and \,$b \in K$
\end{itemize}
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. |
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