Riesz space

In mathematics, a Riesz space, lattice-ordered vector space or vector lattice is a partially ordered vector space where the order structure is a lattice.

Riesz spaces are named after Frigyes Riesz who first defined them in his 1928 paper Sur la décomposition des opérations fonctionelles linéaires.

Riesz spaces have wide ranging applications. They are important in measure theory, in that important results are special cases of results for Riesz Spaces. E.g. the Radon–Nikodym theorem follows as a special case of the Freudenthal spectral theorem. Riesz spaces have also seen application in mathematical economics through the work of Greek-American economist and mathematician Charalambos D. Aliprantis.

Definition 
A Riesz space is defined to be an ordered vector space
for which the ordering is a lattice.

More explicitly, a Riesz space  can be defined to be a vector space endowed with a partial order, , that for any  in , satisfies:
 Translation Invariance:  implies .
 Positive Homogeneity: For any scalar ,  implies .
 For any pair of vectors  in  there exists a supremum (denoted ) in  with respect to the partial order  .

The partial order, together with items 1 and 2, which make it "compatible with the vector space structure", make   an ordered vector space.  Item 3 says that the partial order is a join semilattice.  Because the order is compatible with the vector space structure, one can show that any pair also have an infimum, making  also a meet semilattice, hence a lattice.

Basic properties 
Every Riesz space is a partially ordered vector space, but not every partially ordered vector space is a Riesz space.

Every element  in a Riesz space, , has unique positive and negative parts, written . Then it can be shown that,  and an absolute value can be defined by . Every Riesz space is a distributive lattice and has the Riesz decomposition property.

Order convergence 
There are a number of meaningful non-equivalent ways to define convergence of sequences or nets with respect to the order structure of a Riesz space. A sequence in a Riesz space  is said to converge monotonely if it is a monotone decreasing (resp. increasing) sequence and its infimum (supremum)  exists in  and denoted , (resp. ).

A sequence in a Riesz space  is said to converge in order to  if there exists a monotone converging sequence in  such that .

If  is a positive element of a Riesz space  then a sequence in  is said to converge u-uniformly to  if for any  there exists an  such that  for all .

Subspaces 
The extra structure provided by these spaces provide for distinct kinds of Riesz subspaces. The collection of each kind structure in a Riesz space (e.g. the collection of all ideals) forms a distributive lattice.

Ideals 
A vector subspace  of a Riesz space  is called an ideal if it is solid, meaning if for  and , we have:  implies that . The intersection of an arbitrary collection of ideals is again an ideal, which allows for the definition of a smallest ideal containing some non-empty subset  of , and is called the ideal generated by . An Ideal generated by a singleton is called a principal ideal.

Bands and -Ideals 
A band  in a Riesz space  is defined to be an ideal with the extra property, that for any element  in  for which its absolute value  is the supremum of an arbitrary subset of positive elements in , that  is actually in . -Ideals are defined similarly, with the words 'arbitrary subset' replaced with 'countable subset'. Clearly every band is a -ideal, but the converse is not true in general.

As with ideals, for every non-empty subset  of , there exists a smallest band containing that subset, called the band generated by . A band generated by a singleton is called a principal band.

Disjoint complements 
Two elements  in a Riesz space , are said to be disjoint, written , when  For any subset  of , its disjoint complement  is defined as the set of all elements in , that are disjoint to all elements in . Disjoint complements are always bands, but the converse is not true in general.

Projection bands 
A band  in a Riesz space, is called a projection band, if , meaning every element  in , can be written uniquely as a sum of two elements, , with  in  and  in . There then also exists a positive linear idempotent, or projection, , such that .

The collection of all projection bands in a Riesz space forms a Boolean algebra. Some spaces do not have non-trivial projection bands (e.g. ), so this Boolean algebra may be trivial.

Projection properties 
There are numerous projection properties that Riesz spaces may have. A Riesz space is said to have the (principal) projection property if every (principal) band is a projection band.

The so-called main inclusion theorem relates the following additional properties to the (principal) projection property: A Riesz space is…
 Dedekind Complete (DC) if every nonempty set, bounded above, has a supremum;
 Super Dedekind Complete (SDC) if every nonempty set, bounded above, has a countable subset with identical supremum;
 Dedekind -complete if every countable nonempty set, bounded above, has a supremum; and
 Archimedean property if, for every pair of positive elements  and , there exists an integer  such that .  
Then these properties are related as follows.  SDC implies DC; DC implies both Dedekind -completeness and the projection property; Both Dedekind σ-completeness and the projection property separately imply the principal projection property; and the principal projection property implies the Archimedean property.

None of the reverse implications hold, but Dedekind -completeness and the projection property together imply DC.

Examples 
 The space of continuous real valued functions with compact support on a topological space  with the pointwise partial order defined by  when  for all  in , is a Riesz space. It is Archimedean, but usually does not have the principal projection property unless  satisfies further conditions (e.g. being extremally disconnected).
 Any  with the (almost everywhere) pointwise partial order is a Dedekind complete Riesz space.
 The space  with the lexicographical order is a non-Archimedean Riesz space.

Properties 
 Riesz spaces are lattice ordered groups
 Every Riesz space is a distributive lattice

See also 
 Partially ordered space
 Ordered vector space
 Convex cone

References 

 Bourbaki, Nicolas; Elements of Mathematics: Integration. Chapters 1–6; 
 Riesz, Frigyes; Sur la décomposition des opérations fonctionelles linéaires, Atti congress. internaz. mathematici (Bologna, 1928), 3, Zanichelli  (1930)  pp. 143–148
 
 

Category:Functional analysis
Category:Ordered groups