# Riesz space

vector lattice

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

1) the vector space structure and the partial order are compatible. i.e. from $x, y, z \in X$ and $x < y$ follows that $x+ z < y+ z$ and from $x \in X$, $x > 0$, $\lambda \in \mathbf R$, $\lambda > 0$ follows $\lambda x > 0$;

2) for any two elements $x, y$ there exists $\sup ( x, y) \in X$. In particular, the supremum and infimum of any finite set exist.

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

The space $C[ a, b]$ of real continuous functions with the pointwise order is an example of a Riesz space. For any element $x$ of a Riesz space one can define $x _ {+} = \sup ( x, 0)$, $x _ {-} = \sup (- x, 0)$ and $| x | = x _ {+} + x _ {-}$. It turns out that $x = x _ {+} - x _ {-}$. In Riesz spaces one can introduce two types of convergence of a sequence $\{ x _ {n} \}$. Order convergence, $o$- convergence: $x _ {n} \rightarrow ^ {o} x _ {0}$ if there exist a monotone increasing sequence $\{ y _ {n} \}$ and a monotone decreasing sequence $\{ z _ {n} \}$ such that $y _ {n} \leq x _ {n} \leq z _ {n}$ and $\sup y _ {n} = \inf z _ {n} = x _ {0}$. Relative uniform convergence, $r$- convergence: $x _ {n} \rightarrow ^ {r} x _ {0}$ if there exists an element $u > 0$ such that for any $\epsilon > 0$ there exists an $n _ {0}$ such that $| x _ {n} - x _ {0} | < \epsilon u$ for $n \geq n _ {0}$( $r$- convergence is also called convergence with a regulator). The concepts of $o$- and $r$- convergence have many of the usual properties of convergence of numerical sequences and can be naturally generalized to nets $\{ x _ \alpha \} _ {\alpha \in \mathfrak A } \subset X$.

A Riesz space is called Archimedean if $x, y \in X$ and $nx \leq y$ for $n = 1, 2 , . . .$ imply $x \leq 0$. In Archimedean Riesz spaces, $\lambda _ {n} \rightarrow \lambda _ {0}$ and $x _ {n} \rightarrow ^ {o} x _ {0}$ imply $\lambda _ {n} x _ {n} \rightarrow ^ {o} \lambda _ {0} x _ {0}$( $\lambda _ {n} , \lambda _ {0} \in \mathbf R$, $x _ {n} , x _ {0} \in X$), and $r$- convergence implies $o$- convergence.

How to Cite This Entry:
Riesz space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Riesz_space&oldid=48568
This article was adapted from an original article by V.I. Sobolev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article