Archimedean Property - Definition For Normed Fields

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,

,

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.