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

Preliminaries

If X is an ordered vector space and if S is a subset of X then an element b ∈ X is an upper bound of S if s ≤ b for all s ∈ S.
An element a in X is the least upper bound or supremum of S if it is an upper bound of S and if for any upper bounded of b of S, we have a ≤ b.

Definitions

Prordered vector lattice

A preordered vector lattice is a preordered vector space in which every pair of elements has a supremum.
More explicitly, a preordered vector lattice is vector space endowed with a preorder,, such that for any ∈ :
  1. Translation Invariance: implies.
  2. Positive Homogeneity: For any scalar, implies.
  3. For any pair of vectors in there exists a supremum in with respect to the order.
The preorder, together with items 1 and 2, which make it "compatible with the vector space structure", make an preordered vector space.
Item 3 says that the preorder is a join semilattice.
Because the preorder 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.
A preordered vector space E is a preordered vector lattice if and only if it satisfies any of the following equivalent properties:

  1. For any ∈, their supremum exists in.
  2. For any ∈, their infimum exists in.
  3. For any ∈, their infimum and their supremum exist in.
  4. For any ∈, sup exists.

Riesz space and vector lattices

A Riesz space or a vector lattice is a preordered vector lattice whose preorder is a partial order.
Equivalently, it is an ordered vector space
for which the ordering is a lattice.
Note that many authors required that a vector lattice be a partially ordered vector space while others only require that it be a preordered vector space.
We will henceforth assume that every Riesz space and every vector lattice is an ordered vector space but that a preordered vector lattice is not necessarily partially ordered.
If E is an ordered vector space over with positive whose positive cone C is generating, and if for every x, y ∈ C either or exists, then E is a vector lattice.

Intervals

An order interval in a partially ordered vector space is a convex set of the form =.
In an ordered real vector space, every interval of the form is balanced.
From axioms 1 and 2 above it follows that x,y in and λ in implies λx + y in .
A subset is said to be order bounded if it is contained in some order interval.
An order unit of a preordered vector space is any element x such that the set is absorbing.
The set of all linear functionals on a preordered vector space V that map every order interval into a bounded set is called the order bound dual of V and denoted by Vb
If a space is ordered then its order bound dual is a vector subspace of its algebraic dual.
A subset A of a vector lattice E is called order complete if for every non-empty subset B ⊆ A such that B is order bounded in A, both and exist and are elements of A.
We say that a vector lattice E is order complete is E is an order complete subset of E.

Finite dimesnional Riesz spaces

Finite dimensional vector lattices fall into one of two categories depending on whether or not the lattice is Archimedean ordered.
As with finite dimensional topological vector spaces, finite dimensional vector lattices are thus found to be uninteresting.

Basic properties

Every Riesz space is a partially ordered vector space, but not every partially ordered vector space is a Riesz space.
Note that for any subset A of X, whenever either the supremum or infimum exists.
If and then.
For all a, b, x, and y in a Riesz space X, we have a - inf + b = sup.

Absolute value

For every element x in a Riesz space X, the absolute value of x, denoted by, is defined to be, where this satisfies -|x| ≤ x ≤ |x| and |x| ≥ 0.
For any x and y in X and any real number r, we have and.

Disjointness

We say that two elements x and y in a vector lattice x are lattice disjoint or disjoint if, in which case we write.
Two elements x and y are disjoint if and only if.
If x and y are disjoint then and, where for any element z, and.
We say that two sets A and B are disjoint if a and b are disjoint for all a in A and all b in B, in which case we write.
If A is the singleton set then we will write in place of.
For any set A, we define the disjoint complement to be the set.
Disjoint complements are always bands, but the converse is not true in general.
If A is a subset of X such that exists, and if B is a subset lattice in X that is disjoint from A, then B is a lattice disjoint from.

Representation as a disjoint sum of positive elements

For any x in X, let and, where note that both of these elements are and with.
Then and are disjoint, and is the unique representation of x as the difference of disjoint elements that are.
For all x and y in X, and.
If y ≥ 0 and x ≤ y then x+ ≤ y.
Moreover, if and only if and.
Every Riesz space is a distributive lattice; that is, it has the following equivalent properties: for all x, y, and z in X
  1. x ∧ = ∨
  2. x ∨ = ∧
  3. =.
  4. x'z = y'z and x'z = y'z always imply x=y.
Every Riesz space 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 sequence and its infimum exists in and denoted,.
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 forms a distributive lattice.

Sublattices

If X is a vector lattice then a vector sublattice is a vector subspace F of X such that for all x and y in F, belongs to F.
It can happen that a subspace F of X is a vector lattice under its canonical order but is not a vector sublattice of X.

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.
The intersection of an arbitrary family of bands is again a band.
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.

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, so this Boolean algebra may be trivial.

Completeness

A vector lattice is complete if every subset has both a supremum and an infimum.
A vector lattice is Dedekind complete if each set with an upper bound has a supremum and each set with a lower bound has an infimum.
An order complete, regularly ordered vector lattice whose canonical image in its order bidual is order complete is called minimal and is said to be of minimal type.

Subspaces, quotients, and products

;Sublattices
If M is a vector subspace of a preordered vector space X then the canonical ordering on M induced by Xs positive cone C is the preorder induced by the pointed convex cone C ∩ M, where this cone is proper if C is proper.
A sublattice' of a vector lattice X is a vector subspace M of X such that for all x and y in M, supX belongs to X.
If X = with 0 < p < 1, then the 2-dimensional vector subspace M of X defined by all maps of the form is a vector lattice under the induced order but is not a sublattice of X.
This despite X being an order complete Archimedean ordered topological vector lattice.
Furthermore, there exist vector a vector sublattice N of this space X such that N ∩ C has empty interior in X but no positive linear functional on N can be extended to a positive linear functional on X.
;Quotient lattices
Let M be a vector subspace of an ordered vector space X having positive cone C, let be the canonical projection, and let.
Then is a cone in X/M that induces a canonical preordering on the quotient space X/M.
If is a proper cone in X/M then makes X/M into an ordered vector space.
If M is C-saturated then defines the canonical order of X/M.
Note that provides an example of an ordered vector space where is not a proper cone.
If X is a vector lattice and N is a solid vector subspace of X then defines the canonical order of X/M under which L/M is a vector lattice and the canonical map is a vector lattice homomorphism.
Furthermore, if X is order complete and M is a band in X then X/M is isomorphic with M⟂.
Also, if M is solid then the order topology of X/M is the quotient of the order topology on X.
If X is a topological vector lattice and M is a closed solid sublattice of X then X/L is also a topological vector lattice.
;Product
If S is any set then the space XS of all functions from S into X is canonically ordered by the proper cone.
Suppose that is a family of preordered vector spaces and that the positive cone of is.
Then is a pointed convex cone in, which determines a canonical ordering on ;
C is a proper cone if all are proper cones.
;Algebraic direct sum
The algebraic direct sum of is a vector subspace of that is given the canonical subspace ordering inherited from.
If X1,..., Xn are ordered vector subspaces of an ordered vector space X then X is the ordered direct sum of these subspaces if the canonical algebraic isomorphism of X onto is an order isomorphism.

Spaces of linear maps

A cone C in a vector space X is said to be generating if C − C is equal to the whole vector space.
If X and W are two non-trivial ordered vector spaces with respective positive cones P and Q, then P is generating in X if and only if the set is a proper cone in L, which is the space of all linear maps from X into W.
In this case the ordering defined by C is called the canonical ordering of L.
More generally, if M is any vector subspace of L such that C ∩ M is a proper cone, the ordering defined by C ∩ M is called the canonical ordering of M.
A linear map u between two preordered vector spaces X and Y with respective positive cones C and D is called positive if u ⊆ D.
If X and Y are vector lattices with Y order complete and if H is the set of all positive linear maps from X into Y then the subspace M := H - H of L is an order complete vector lattice under its canonical order;
furthermore, M contains exactly those linear maps that map order intervals of X into order intervals of Y.

Positive functionals and the order dual

A linear function f on a preordered vector space is called positive if x ≥ 0 implies f ≥ 0.
The set of all positive linear forms on a vector space, denoted by, is a cone equal to the polar of −C.
The order dual of an ordered vector space X is the set, denoted by, defined by.
Although, there do exist ordered vector spaces for which set equality does not hold.

Vector lattice homomorphism

Suppose that X and Y are preordered vector lattices with positive cones C and D and let u be a map from X into Y.
Then u is a preordered vector lattice homomorphism if u is linear and if any one of the following equivalent conditions hold:

  1. u preserves the lattice operations
  2. u = sup for all x, y ∈ X
  3. u = inf for all x, y ∈ X
  4. u = sup for all x ∈ X
  5. 0 = inf for all x ∈ X
  6. u = D and u−1 is a solid subset of X.
  7. if x ≥ 0 then u ≥ 0.
  8. u is order preserving.
A pre-ordered vector lattice homomorphism that is bijective is a pre-ordered vector lattice isomorphism.
A pre-ordered vector lattice homomorphism between two Riesz spaces is called a vector lattice homomorphism;
if it is also bijective, then it is called a vector lattice isomorphism.
If u is a non-0 linear functional on a vector lattice X with positive cone C then the following are equivalent:

  1. u : X is a surjective vector lattice homomorphism.
  2. 0 = inf for all x ∈ X
  3. u ≥ 0 and u−1 is a solid hyperplane in X.
  4. u generates an extreme ray of the cone C* in X*
Recall that an extreme ray' of the cone C is a set where x ∈C, x is non-0, and if y ∈C is such that x - y ∈C then y = s x for some s such that 0 ≤ s ≤ 1.
A vector lattice homomorphism from X into Y is a topological homomorphism when X and Y are given their respective order topologies.

Projection properties

There are numerous projection properties that Riesz spaces may have. A Riesz space is said to have the projection property if every band is a projection band.
The so-called main inclusion theorem relates the following additional properties to the projection property: A Riesz space is…
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