# Riesz space

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vector lattice

A real partially ordered vector space (cf. Partially ordered set; Vector space) in which

1) the vector space structure and the partial order are compatible. i.e. from and follows that and from , , , follows ;

2) for any two elements there exists . In particular, the supremum and infimum of any finite set exist.

In Soviet scientific literature Riesz spaces are usually called -lineals. Such spaces were first introduced by F. Riesz in 1928.

The space of real continuous functions with the pointwise order is an example of a Riesz space. For any element of a Riesz space one can define , and . It turns out that . In Riesz spaces one can introduce two types of convergence of a sequence . Order convergence, -convergence: if there exist a monotone increasing sequence and a monotone decreasing sequence such that and . Relative uniform convergence, -convergence: if there exists an element such that for any there exists an such that for ( -convergence is also called convergence with a regulator). The concepts of - and -convergence have many of the usual properties of convergence of numerical sequences and can be naturally generalized to nets .

A Riesz space is called Archimedean if and for imply . In Archimedean Riesz spaces, and imply ( , ), and -convergence implies -convergence.

How to Cite This Entry:
Riesz space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Riesz_space&oldid=17299
This article was adapted from an original article by V.I. Sobolev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article