Difference between revisions of "Ultrametric"
From Encyclopedia of Mathematics
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| − | * Natarajan, P. N. "An introduction to ultrametric summability theory". SpringerBriefs in Mathematics. Springer (2014) ISBN 978-81-322-1646-9 {{ZBL|1284.40001}} | + | * 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}} | + | * 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}} |
Latest revision as of 17:38, 13 October 2023
2020 Mathematics Subject Classification: Primary: 54E35,16W60 [MSN][ZBL]
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=40921
Ultrametric. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Ultrametric&oldid=40921