and the following are equivalent: satisfies the ultrametric inequality for all , , in . is bounded in R. for every . for all . for all . An absolute value which satisfies any (hence all) of the above conditions is said to be non-Archimedean, otherwise it is said to be Archimedean. Vector spaces Again the fundamental properties of the absolute value for real numbers can be used, with a slight modification, to generalise the notion to an arbitrary vector space. A real-valued function on a vector space over a field , represented as , is called an absolute value, but more usually a norm, if it satisfies the following axioms: For all in , and , in , {| cellpadding=10 |- | |Non-negativity |- | |Positive-definiteness |- | |Positive homogeneity or positive scalability |- | |Subadditivity or the triangle inequality |} The norm of a vector is also called its length or magnitude. In the case of Euclidean space , the function defined by is a norm called the Euclidean norm. When the real numbers are considered as the one-dimensional vector space , the absolute value is a norm, and is the -norm (see Lp space) for any . In fact the absolute value is the "only" norm on , in the sense that, for every norm on , . The complex absolute value is a special case of the norm in an inner product space, which is identical to the Euclidean norm when the complex plane is identified as the Euclidean plane . Composition algebras Every composition algebra A has an involution x → x* called its conjugation. The product in A of an element x and its conjugate x* is written N(x) = x x* and called the norm of x. The real numbers , complex numbers , and quaternions are all composition algebras with norms given by definite quadratic forms. The absolute value in these division algebras is given by the square root of the composition algebra norm. In general the norm of a composition algebra may be a quadratic form that is not definite and has null vectors. However, as in the case of division algebras, when an element x has a non-zero norm, then x has a multiplicative inverse given by x*/N(x). See also Least absolute values Notes References Bartle; Sherbert; Introduction to real analysis (4th ed.), John Wiley & Sons, 2011 . Nahin,