Absolute Value¶
Recall (doc/postdoc). Not a first course.
Job: archimedean absolute value on \(\mathbb{R}\); prototype of norms and valuations.
Core¶
\[
|x|=\begin{cases}x&x\ge0\\ -x&x<0\end{cases}
=\max(x,-x),\qquad |x|=\sqrt{x^2}
\]
| Property | Formula |
|---|---|
| Positive definite | $ |
| Multiplicative | $ |
| Triangle | $ |
| Reverse triangle | $ |
Equations: \(|x|=c\) (\(c>0\)) \(\Rightarrow x=\pm c\); \(|x|<c\Rightarrow -c<x<c\); \(|x|>c\Rightarrow x<-c\) or \(x>c\).
Metric: \(d(x,y)=|x-y|\) on \(\mathbb{R}\); completion story → \(\mathbb{R}\) from \(\mathbb{Q}\).
Used later for¶
- Metric spaces: general \(d\)
- Normed spaces: \(\|x\|\)
- Number theory: \(p\)-adic \(|\cdot|_p\), Ostrowski
- Complex analysis: \(|z|=\sqrt{z\bar z}\)
Check¶
Which absolute values (up to equivalence) exist on \(\mathbb{Q}\)?
Ostrowski: archimedean \(|\cdot|_\infty\) and non-archimedean \(|\cdot|_p\) for each prime \(p\) (and trivial).