# Difference between revisions of "Ultrametric"

From Encyclopedia of Mathematics

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## Revision as of 20:32, 9 April 2017

An ultrametric is a metric $d$ on a set $X$ satisfying the *strong triangle inequality*: $d(x,y) \le \max\{d(x,z), d(z,y)\}$. An ultrametric topology is one induced by an ultrametric.

An ultrametric valuation $\Vert{\cdot}\Vert$ on a ring $R$ similarly satisfies the condition: $\Vert x+y \Vert \le \max\{\Vert x \Vert, \Vert y \Vert\}$. An example is the $p$-adic valuation.

The term **non-Archimedean** is also used.

#### References

- Natarajan, P. N. "An introduction to ultrametric summability theory". SpringerBriefs in Mathematics. Springer (2014) ISBN 978-81-322-1646-9 Zbl 1284.40001
- Schikhof, W.H. "Ultrametric calculus. An introduction to $p$-adic analysis". Cambridge Studies in Advanced Mathematics
**4**Cambridge University Press (1984) ISBN 0-521-24234-7 Zbl 0553.26006

**How to Cite This Entry:**

Ultrametric.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Ultrametric&oldid=40919