# 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 E can be defined to be a vector space endowed with a partial order, ≤, that for any *x*, *y*, *z* in E, satisfies:

- Translation Invariance:
*x*≤*y*implies*x*+*z*≤*y*+*z*. - Positive Homogeneity: For any scalar 0 ≤
*α*,*x*≤*y*implies*αx*≤*αy*. - For any pair of vectors
*x*,*y*in E there exists a supremum (denoted*x*∨*y*) in E 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 E 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 E 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 f in a Riesz space, E, has unique positive and negative parts, written *f*^{ ±} = ±*f* ∨ 0. Then it can be shown that, *f* = *f*^{ +} − *f*^{ −} and an absolute value can be defined by | *f* | = *f*^{ +} + *f*^{ −}. 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 {*x _{n}*} in a Riesz space E is said to

**converge monotonely**if it is a monotone decreasing (resp. increasing) sequence and its infimum (supremum) x exists in E and denoted

*x*↓

_{n}*x*, (resp.

*x*↑

_{n}*x*).

A sequence {*x _{n}*} in a Riesz space E is said to

**converge in order**to x if there exists a monotone converging sequence {

*p*} in E such that |

_{n}*x*−

_{n}*x*| <

*p*↓ 0.

_{n}If u is a positive element of a Riesz space E then a sequence {*x _{n}*} in E is said to

**converge u-uniformly**to x if for any

*ε*> 0 there exists an N such that |

*x*−

_{n}*x*| <

*εu*for all

*n*>

*N*.

## 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 I of a Riesz space E is called an *ideal* if it is *solid*, meaning if for *f* ∈ *I* and *g* ∈ *E*, we have: |*g*| ≤ | *f* | implies that *g* ∈ *I*. 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 A of E, and is called the ideal *generated* by A. An Ideal generated by a singleton is called a **principal ideal**.

### Bands and σ-Ideals

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

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

### Disjoint complements

Two elements *f*, *g* in a Riesz space E, are said to be **disjoint**, written *f* ⊥ *g*, when | *f* | ∧ |*g*| = 0. For any subset A of E, its disjoint complement *A ^{d}* is defined as the set of all elements in E, that are disjoint to all elements in A. Disjoint complements are always bands, but the converse is not true in general.

### Projection bands

A band B in a Riesz space, is called a *projection band*, if *E* = *B* ⊕ *B ^{d}*, meaning every element f in E, can be written uniquely as a sum of two elements,

*f*=

*u*+

*v*, with u in B and v in

*B*. There then also exists a positive linear idempotent, or

^{d}*projection*,

*P*:

_{B}*E*→

*E*, such that

*P*(

_{B}*f*) =

*u*.

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. *C*([0, 1])), 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:[1] 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 σ-complete if every countable nonempty set, bounded above, has a supremum; and
- Archimedean property if, for every pair of positive elements x and y, there exists an integer n such that
*nx*≥*y*.

Then these properties are related as follows. SDC implies DC; DC implies both Dedekind σ-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 σ-completeness and the projection property together imply DC.

## Examples

- The space of continuous real valued functions with compact support on a topological space X with the pointwise partial order defined by
*f*≤*g*when*f*(*x*) ≤*g*(*x*) for all x in X, is a Riesz space. It is Archimedean, but usually does not have the principal projection property unless X satisfies further conditions (e.g. being extremally disconnected). - Any
*L*with the (almost everywhere) pointwise partial order is a Dedekind complete Riesz space.^{p} - The space
**R**^{2}with the lexicographical order is a non-Archimedean Riesz space.

## Properties

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

## References

- Luxemburg, W.A.J.; Zaanen, A.C. (1971).
*Riesz Spaces : Vol. 1*. London: North Holland. pp. 122–138. ISBN 0720424518. Retrieved 8 January 2018.

- Bourbaki, Nicolas; Elements of Mathematics: Integration. Chapters 1–6; ISBN 3-540-41129-1
- Riesz, Frigyes; Sur la décomposition des opérations fonctionelles linéaires, Atti congress. internaz. mathematici (Bologna, 1928), 3, Zanichelli (1930) pp. 143–148
- Sobolev, V. I. (2001), "Riesz space",
*Encyclopædia of Mathematics*, Springer, ISBN 978-1-4020-0609-8 - Zaanen, Adriaan C. (1996),
*Introduction to Operator Theory in Riesz spaces*, Springer, ISBN 3-540-61989-5