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. For example, 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

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Preliminaries

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If   is an ordered vector space (which by definition is a vector space over the reals) and if   is a subset of   then an element   is an upper bound (resp. lower bound) of   if   (resp.  ) for all   An element   in   is the least upper bound or supremum (resp. greater lower bound or infimum) of   if it is an upper bound (resp. a lower bound) of   and if for any upper bound (resp. any lower bound)   of     (resp.  ).

Definitions

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Preordered vector lattice

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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   there exists a supremum (denoted  ) in   with respect to the order  

The preorder, together with items 1 and 2, which make it "compatible with the vector space structure", make   a 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   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     exists in  [1]

Riesz space and vector lattices

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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 (rather than merely a preordered 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   is an ordered vector space over   whose positive cone   (the elements  ) is generating (that is, such that  ), and if for every   either   or   exists, then   is a vector lattice.[2]

Intervals

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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.[3] From axioms 1 and 2 above it follows that   and   implies   A subset is said to be order bounded if it is contained in some order interval.[3] An order unit of a preordered vector space is any element   such that the set   is absorbing.[3]

The set of all linear functionals on a preordered vector space   that map every order interval into a bounded set is called the order bound dual of   and denoted by  [3] If a space is ordered then its order bound dual is a vector subspace of its algebraic dual.

A subset   of a vector lattice   is called order complete if for every non-empty subset   such that   is order bounded in   both   and   exist and are elements of   We say that a vector lattice   is order complete if   is an order complete subset of  [4]

Classification

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Finite-dimensional Riesz spaces are entirely classified by the Archimedean property:

Theorem:[5] Suppose that   is a vector lattice of finite-dimension   If   is Archimedean ordered then it is (a vector lattice) isomorphic to   under its canonical order. Otherwise, there exists an integer   satisfying   such that   is isomorphic to   where   has its canonical order,   is   with the lexicographical order, and the product of these two spaces has the canonical product order.

The same result does not hold in infinite dimensions. For an example due to Kaplansky, consider the vector space V of functions on [0,1] that are continuous except at finitely many points, where they have a pole of second order. This space is lattice-ordered by the usual pointwise comparison, but cannot be written as κ for any cardinal κ.[6] On the other hand, epi-mono factorization in the category of -vector spaces also applies to Riesz spaces: every lattice-ordered vector space injects into a quotient of κ by a solid subspace.[7]

Basic properties

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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   of     whenever either the supremum or infimum exists (in which case they both exist).[2] If   and   then  [2] For all   in a Riesz space    [4]

Absolute value

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For every element   in a Riesz space   the absolute value of   denoted by   is defined to be  [4] where this satisfies   and   For any   and any real number   we have   and  [4]

Disjointness

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Two elements   in a vector lattice   are said to be lattice disjoint or disjoint if   in which case we write   Two elements   are disjoint if and only if   If   are disjoint then   and   where for any element     and   We say that two sets   and   are disjoint if   and   are disjoint for all   and all   in which case we write  [2] If   is the singleton set   then we will write   in place of   For any set   we define the disjoint complement to be the set  [2] Disjoint complements are always bands, but the converse is not true in general. If   is a subset of   such that   exists, and if   is a subset lattice in   that is disjoint from   then   is a lattice disjoint from  [2]

Representation as a disjoint sum of positive elements

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For any   let   and   where note that both of these elements are   and   with   Then   and   are disjoint, and   is the unique representation of   as the difference of disjoint elements that are  [2] For all     and  [2] If   and   then   Moreover,   if and only if   and  [2]

Every Riesz space is a distributive lattice; that is, it has the following equivalent[Note 1] properties:[8] for all  

  1.  
  2.  
  3.  
  4.  and   always imply  

Every Riesz space has the Riesz decomposition property.

Order convergence

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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

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

Sublattices

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If   is a vector lattice then a vector sublattice is a vector subspace   of   such that for all     belongs to   (where this supremum is taken in  ).[4] It can happen that a subspace   of   is a vector lattice under its canonical order but is not a vector sublattice of  [4]

Ideals

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A vector subspace   of a Riesz space   is called an ideal if it is solid, meaning if for   and     implies that  [4] 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 σ-Ideals

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A band   in a Riesz space   is defined to be an ideal with the extra property, that for any element   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

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A band   in a Riesz space, is called a projection band, if   meaning every element   can be written uniquely as a sum of two elements,   with   and   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 (for example,  ), so this Boolean algebra may be trivial.

Completeness

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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.[4]

Subspaces, quotients, and products

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Sublattices

If   is a vector subspace of a preordered vector space   then the canonical ordering on   induced by  's positive cone   is the preorder induced by the pointed convex cone   where this cone is proper if   is proper (that is, if  ).[3]

A sublattice of a vector lattice   is a vector subspace   of   such that for all     belongs to   (importantly, note that this supremum is taken in   and not in  ).[3] If   with   then the 2-dimensional vector subspace   of   defined by all maps of the form   (where  ) is a vector lattice under the induced order but is not a sublattice of  [5] This despite   being an order complete Archimedean ordered topological vector lattice. Furthermore, there exist vector a vector sublattice   of this space   such that   has empty interior in   but no positive linear functional on   can be extended to a positive linear functional on  [5]

Quotient lattices

Let   be a vector subspace of an ordered vector space   having positive cone   let   be the canonical projection, and let   Then   is a cone in   that induces a canonical preordering on the quotient space   If   is a proper cone in   then   makes   into an ordered vector space.[3] If   is  -saturated then   defines the canonical order of  [5] Note that   provides an example of an ordered vector space where   is not a proper cone.

If   is a vector lattice and   is a solid vector subspace of   then   defines the canonical order of   under which   is a vector lattice and the canonical map   is a vector lattice homomorphism. Furthermore, if   is order complete and   is a band in   then   is isomorphic with  [5] Also, if   is solid then the order topology of   is the quotient of the order topology on  [5]

If   is a topological vector lattice and   is a closed solid sublattice of   then   is also a topological vector lattice.[5]

Product

If   is any set then the space   of all functions from   into   is canonically ordered by the proper cone  [3]

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  ;   is a proper cone if all   are proper cones.[3]

Algebraic direct sum

The algebraic direct sum   of   is a vector subspace of   that is given the canonical subspace ordering inherited from  [3] If   are ordered vector subspaces of an ordered vector space   then   is the ordered direct sum of these subspaces if the canonical algebraic isomorphism of   onto   (with the canonical product order) is an order isomorphism.[3]

Spaces of linear maps

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A cone   in a vector space   is said to be generating if   is equal to the whole vector space.[3] If   and   are two non-trivial ordered vector spaces with respective positive cones   and   then   is generating in   if and only if the set   is a proper cone in   which is the space of all linear maps from   into   In this case the ordering defined by   is called the canonical ordering of  [3] More generally, if   is any vector subspace of   such that   is a proper cone, the ordering defined by   is called the canonical ordering of  [3]

A linear map   between two preordered vector spaces   and   with respective positive cones   and   is called positive if   If   and   are vector lattices with   order complete and if   is the set of all positive linear maps from   into   then the subspace   of   is an order complete vector lattice under its canonical order; furthermore,   contains exactly those linear maps that map order intervals of   into order intervals of  [5]

Positive functionals and the order dual

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A linear function   on a preordered vector space is called positive if   implies   The set of all positive linear forms on a vector space, denoted by   is a cone equal to the polar of   The order dual of an ordered vector space   is the set, denoted by   defined by   Although   there do exist ordered vector spaces for which set equality does not hold.[3]

Vector lattice homomorphism

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Suppose that   and   are preordered vector lattices with positive cones   and   and let   be a map. Then   is a preordered vector lattice homomorphism if   is linear and if any one of the following equivalent conditions hold:[9][5]

  1.   preserves the lattice operations
  2.   for all  
  3.   for all  
  4.   for all  
  5.   for all  
  6.   and   is a solid subset of  [5]
  7. if   then  [1]
  8.   is order preserving.[1]

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   is a non-zero linear functional on a vector lattice   with positive cone   then the following are equivalent:

  1.   is a surjective vector lattice homomorphism.
  2.   for all  
  3.   and   is a solid hyperplane in  
  4.   generates an extreme ray of the cone   in  

An extreme ray of the cone   is a set   where     is non-zero, and if   is such that   then   for some   such that  [9]

A vector lattice homomorphism from   into   is a topological homomorphism when   and   are given their respective order topologies.[5]

Projection properties

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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:[10] 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  , whenever the inequality   holds for all integers  ,  .

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

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  • The space of continuous real valued functions with compact support on a topological space   with the pointwise partial order defined by   when   for all   is a Riesz space. It is Archimedean, but usually does not have the principal projection property unless   satisfies further conditions (for example, being extremally disconnected).
  • Any   space 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

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See also

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Notes

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  1. ^ The conditions are equivalent only when they apply to all triples in a lattice. There are elements in (for example) N5 that satisfy the first equation but not the second.

References

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  1. ^ a b c Narici & Beckenstein 2011, pp. 139–153.
  2. ^ a b c d e f g h i Schaefer & Wolff 1999, pp. 74–78.
  3. ^ a b c d e f g h i j k l m n o Schaefer & Wolff 1999, pp. 205–209.
  4. ^ a b c d e f g h Schaefer & Wolff 1999, pp. 204–214.
  5. ^ a b c d e f g h i j k Schaefer & Wolff 1999, pp. 250–257.
  6. ^ Birkhoff 1967, p. 240.
  7. ^ Fremlin, Measure Theory, claim 352L.
  8. ^ Birkhoff, Garrett (1967). Lattice Theory. Colloquium Publications (3rd ed.). American Mathematical Society. p. 11. ISBN 0-8218-1025-1. §6, Theorem 9
  9. ^ a b Schaefer & Wolff 1999, pp. 205–214.
  10. ^ Luxemburg, W.A.J.; Zaanen, A.C. (1971). Riesz Spaces : Vol. 1. London: North Holland. pp. 122–138. ISBN 0720424518. Retrieved 8 January 2018.

Bibliography

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