In mathematics, a Riesz space, latticeordered 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 wideranging 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 GreekAmerican economist and mathematician Charalambos D. Aliprantis.
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. ).
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 :
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:
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]}
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 nonempty 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]}
Finitedimensional Riesz spaces are entirely classified by the Archimedean property:
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 latticeordered by the usual pointwise comparison, but cannot be written as ℝ^{κ} for any cardinal κ.^{[6]} On the other hand, epimono factorization in the category of ℝvector spaces also applies to Riesz spaces: every latticeordered vector space injects into a quotient of ℝ^{κ} by a solid subspace.^{[7]}
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]}
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]}
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]}
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
Every Riesz space has the Riesz decomposition property.
There are a number of meaningful nonequivalent 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 uuniformly to if for any there exists an such that for all
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.
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]}
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 nonempty subset of and is called the ideal generated by An Ideal generated by a singleton is called a principal ideal.
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 nonempty 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.
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 nontrivial projection bands (for example, ), so this Boolean algebra may be trivial.
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]}
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 2dimensional 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]}
A cone in a vector space is said to be generating if is equal to the whole vector space.^{[3]} If and are two nontrivial 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]}
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]}
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]}
A preordered vector lattice homomorphism that is bijective is a preordered vector lattice isomorphism.
A preordered 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 nonzero linear functional on a vector lattice with positive cone then the following are equivalent:
An extreme ray of the cone is a set where is nonzero, 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]}
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 socalled main inclusion theorem relates the following additional properties to the (principal) projection property:^{[10]} 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 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.
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