| Version 18 |
Version 17 |
| Theorem: A normed field is isomorphic either to the field $\mathbb{R}$ of real numbers or to the field $\mathbb{C}$ of complex numbers. |
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: |
Definition: The {\em normed field} means a field $K$ having as its subfield a field $R$ isomorphic to $\mathbb{R}$ and satisfying the following: |
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There is a mapping $\|.\|$ from $K$ to the set of non-negative reals such that
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There is a mapping $\|\cdot\|$ from $K$ to the set of non-negative reals such that
|
| \begin{itemize} |
\begin{itemize} |
| \item $\|a\| = 0$ iff $a = 0$ |
\item $\|a\| = 0$ iff $a = 0$ |
| \item $\|ab\| \le \|a\|\cdot\|b\|$ |
\item $\|ab\| \le \|a\|\cdot\|b\|$ |
| \item $\|a+b\| \le \|a\|+\|b\|$ |
\item $\|a+b\| \le \|a\|+\|b\|$ |
| \item $\|ab\| = |a|\cdot\|b\|$ when $a \in R$ and $b \in K$ |
\item $\|ab\| = |a|\cdot\|b\|$ when $a \in R$ and $b \in K$ |
| \end{itemize} |
\end{itemize} |
