Abstract
Let K be a field. An absolute value v on K is a real-valued function x ↦ |x| v on K satisfying the following three properties:
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AV 1
We have |x| v ≧ 0 for all x ∊ K, and |x| v = 0 if and only if x = 0.
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AV 2
For all x, y ∊ K, we have |xy| v = |x| v |y| v .
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AV 3
For all x, y ∊ K, we have |x + y| v ≦ |x| v + |y| v .
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Bibliography
S. Lang, Real and Functional Analysis, Springer Verlag, 1993
S. Lang, Undergraduate Algebra, Second Edition, Springer Verlag, 1990
S. Lang, Cyclotomic Fields I and II, Springer Verlag 1990 (combined from the first editions, 1978 and 1980)
C. Rickart, Banach Algebras, Van Nostrand (1960), Theorems 1.7.1 and 4.2.2.
W. Rudin, Functional Analysis, McGraw Hill (1973) Theorems 10.14 and 11.18.
J. P. Serre, Endomorphismes complêtement continus des espaces de Banach p-adiques, Pub. Math. IHES 12 (1962), pp. 69–85.
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© 2002 Springer Science+Business Media New York
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Lang, S. (2002). Absolute Values. In: Algebra. Graduate Texts in Mathematics, vol 211. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-0041-0_12
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DOI: https://doi.org/10.1007/978-1-4613-0041-0_12
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