Definition For Normed Fields
The qualifier "Archimedean" is also formulated in the theory of rank one valued fields and normed spaces over rank one valued fields as follows. Let F be a field endowed with an absolute value function, i.e., a function which associates the real number 0 with the field element 0 and associates a positive real number with each non-zero and satisfies and . Then, F is said to be Archimedean if for any non-zero there exists a natural number n such that
Similarly, a normed space is Archimedean if a sum of terms, each equal to a non-zero vector, has norm greater than one for sufficiently large . A field with an absolute value or a normed space is either Archimedean or satisfies the stronger condition, referred to as the ultrametric triangle inequality,
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respectively. A field or normed space satisfying the ultrametric triangle inequality is called non-Archimedean.
The concept of a non-Archimedean normed linear space was introduced by A. F. Monna.
Read more about this topic: Archimedean Property
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