Riesz spaces

Abstract

A Riesz space is a real vector space equipped with a partial order satisfying the following properties. Inequalities are preserved by adding the same vector to each side, or by multiplying both sides by the same positive scalar. Each pair {x, y} of vectors has a supremum or least upper bound, denoted x V y. Thus Riesz spaces mimic some of the order properties possessed by the real numbers. However, the real numbers possess other properties not shared by all Riesz spaces, such as order completeness and the Archimedean property. To further complicate matters, the norm of a real number coincides with its absolute value. In more general normed Riesz spaces the norm and absolute value are different.