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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: |
2004-02-27 05:07:49 |
| Classification: |
msc:12J05 |
| Keywords: |
real numbers, complex numbers |
Preamble:
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Content:
Theorem: A normed field is isomorphic either to the field $\mathbb{R}$ of real numbers or to the field $\mathbb{C}$ of complex numbers.
Definition: The {\em normed field} means a field $K$ having as its subfield a field $R$ 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 $\|ab\| \le \|a\|\cdot\|b\|$
\item $\|a+b\| \le \|a\|+\|b\|$
\item $\|ab\| = |a|\cdot\|b\|$ when $a \in R$ and $b \in K$
\end{itemize} |
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