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.

Read more about this topic:  Archimedean Property

Famous quotes containing the words definition and/or fields:

    Beauty, like all other qualities presented to human experience, is relative; and the definition of it becomes unmeaning and useless in proportion to its abstractness. To define beauty not in the most abstract, but in the most concrete terms possible, not to find a universal formula for it, but the formula which expresses most adequately this or that special manifestation of it, is the aim of the true student of aesthetics.
    Walter Pater (1839–1894)

    I don’t pop my cork for ev’ry guy I see.
    —Dorothy Fields (1904–1974)