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, $\leq$ , that for any $x, y, z$ in $E$ , satisfies:

Translation Invariance: $x \leq y$ implies $x + z \leq y + z$ .
Positive Homogeneity: For any scalar $0 \leq α$ , $x \leq y$ implies $αx \leq αy$ .
For any pair of vectors $x, y$ in $E$ there exists a supremum (denoted $x \lor y$ ) in $E$ with respect to the partial order $(\leq)$ .

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 $f \pm = \pm f \lor 0$ . 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 ${p n}$ in $E$ such that $| x n - x | < p n ↓ 0$ .

If $u$ is a positive element 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 $f \in I$ and $g \in E$ , we have: $| g | \leq | f |$ implies that $g \in I$ . 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 $| f | \land | g | = 0.$ 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 $E = B \oplus B d$ , 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 $B d$ . There then also exists a positive linear idempotent, or projection, $P B : E \to E$ , 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 $nx \geq y$ .

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 $f \leq g$ when $f (x) \leq g (x)$ 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 $L p$ with the (almost everywhere) pointwise partial order is a Dedekind complete Riesz space.
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

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

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