Ordered vector space

In mathematics, an ordered vector space or partially ordered vector space is a vector space equipped with a partial order that is compatible with the vector space operations.
Definition.
Given a vector space formula_1 over the real numbers formula_2 and a preorder formula_3 on the set formula_4 the pair formula_5 is called a preordered vector space and we say that the preorder formula_3 is compatible with the vector space structure of formula_1 and call formula_3 a vector preorder on formula_1 if for all formula_10 and formula_11 with formula_12 the following two axioms are satisfied
If formula_3 is a partial order compatible with the vector space structure of formula_1 then formula_5 is called an ordered vector space and formula_3 is called a vector partial order on formula_21 
The two axioms imply that translations and positive homotheties are automorphisms of the order structure and the mapping formula_22 is an isomorphism to the dual order structure. Ordered vector spaces are ordered groups under their addition operation.
Note that formula_13 if and only if formula_24
Positive cones and their equivalence to orderings.
A subset formula_25 of a vector space formula_1 is called a cone if for all real formula_27 formula_28 A cone is called pointed if it contains the origin. A cone formula_25 is convex if and only if formula_30 The intersection of any non-empty family of cones (resp. convex cones) is again a cone (resp. convex cone); 
the same is true of the union of an increasing (under set inclusion) family of cones (resp. convex cones). A cone formula_25 in a vector space formula_1 is said to be generating if formula_33 
A positive cone is generating if and only if it is a directed set under formula_34 
Given a preordered vector space formula_4 the subset formula_36 of all elements formula_37 in formula_5 satisfying formula_39 is a pointed convex cone with vertex formula_40 (that is, it contains formula_40) called the positive cone of formula_1 and denoted by formula_43 
The elements of the positive cone are called positive. 
If formula_37 and formula_45 are elements of a preordered vector space formula_46 then formula_13 if and only if formula_48
Given any pointed convex cone formula_25 with vertex formula_50 one may define a preorder formula_3 on formula_1 that is compatible with the vector space structure of formula_1 by declaring for all formula_54 that formula_13 if and only if formula_56 
the positive cone of this resulting preordered vector space is formula_57 
There is thus a one-to-one correspondence between pointed convex cones with vertex formula_40 and vector preorders on formula_21 
If formula_1 is preordered then we may form an equivalence relation on formula_1 by defining formula_37 is equivalent to formula_45 if and only if formula_13 and formula_65 
if formula_66 is the equivalence class containing the origin then formula_66 is a vector subspace of formula_1 and formula_69 is an ordered vector space under the relation: formula_70 if and only there exist formula_71 and formula_72 such that formula_73 
A subset of formula_25 of a vector space formula_1 is called a proper cone if it is a convex cone of vertex formula_40 satisfying formula_77 
Explicitly, formula_25 is a proper cone if (1) formula_79 (2) formula_80 for all formula_27 and (3) formula_77 
The intersection of any non-empty family of proper cones is again a proper cone. Each proper cone formula_25 in a real vector space induces an order on the vector space by defining formula_13 if and only if formula_85 and furthermore, the positive cone of this ordered vector space will be formula_57 Therefore, there exists a one-to-one correspondence between the proper convex cones of formula_1 and the vector partial orders on formula_21 
By a total vector ordering on formula_1 we mean a total order on formula_1 that is compatible with the vector space structure of formula_21 
The family of total vector orderings on a vector space formula_1 is in one-to-one correspondence with the family of all proper cones that are maximal under set inclusion. 
A total vector ordering "cannot" be Archimedean if its dimension, when considered as a vector space over the reals, is greater than 1. 
If formula_93 and formula_94 are two orderings of a vector space with positive cones formula_95 and formula_96 respectively, then we say that formula_93 is finer than formula_94 if formula_99
Examples.
The real numbers with the usual ordering form a totally ordered vector space. For all integers formula_100 the Euclidean space formula_101 considered as a vector space over the reals with the lexicographic ordering forms a preordered vector space whose order is Archimedean if and only if formula_102. 
Pointwise order.
If formula_94 is any set and if formula_1 is a vector space (over the reals) of real-valued functions on formula_105 then the pointwise order on formula_1 is given by, for all formula_107 formula_108 if and only if formula_109 for all formula_110 
Spaces that are typically assigned this order include:
The space formula_121 of all measurable almost-everywhere bounded real-valued maps on formula_122 where the preorder is defined for all formula_123 by formula_108 if and only if formula_109 almost everywhere.
Intervals and the order bound dual.
An order interval in a preordered vector space is set of the form 
formula_126
From axioms 1 and 2 above it follows that formula_127 and formula_128 implies formula_129 belongs to formula_130 
thus these order intervals are convex. 
A subset is said to be order bounded if it is contained in some order interval. 
In a preordered real vector space, if for formula_39 then the interval of the form formula_132 is balanced. 
An order unit of a preordered vector space is any element formula_37 such that the set formula_132 is absorbing. 
The set of all linear functionals on a preordered vector space formula_1 that map every order interval into a bounded set is called the order bound dual of formula_1 and denoted by formula_137 
If a space is ordered then its order bound dual is a vector subspace of its algebraic dual. 
A subset formula_138 of an ordered vector space formula_1 is called order complete if for every non-empty subset formula_140 such that formula_141 is order bounded in formula_142 both formula_143 and formula_144 exist and are elements of formula_145 We say that an ordered vector space formula_1 is order complete is formula_1 is an order complete subset of formula_21
Examples.
If formula_5 is a preordered vector space over the reals with order unit formula_150 then the map formula_151 is a sublinear functional.
Properties.
If formula_1 is a preordered vector space then for all formula_54 
Spaces of linear maps.
A cone formula_25 is said to be generating if formula_178 is equal to the whole vector space. 
If formula_1 and formula_180 are two non-trivial ordered vector spaces with respective positive cones formula_95 and formula_96 then formula_95 is generating in formula_1 if and only if the set formula_185 is a proper cone in formula_186 which is the space of all linear maps from formula_1 into formula_188 
In this case, the ordering defined by formula_25 is called the canonical ordering of formula_190 
More generally, if formula_191 is any vector subspace of formula_192 such that formula_193 is a proper cone, the ordering defined by formula_193 is called the canonical ordering of formula_195 
Positive functionals and the order dual.
A linear function formula_196 on a preordered vector space is called positive if it satisfies either of the following equivalent conditions: 
The set of all positive linear forms on a vector space with positive cone formula_201 called the dual cone and denoted by formula_202 is a cone equal to the polar of formula_203 
The preorder induced by the dual cone on the space of linear functionals on formula_1 is called the . 
The order dual of an ordered vector space formula_1 is the set, denoted by formula_206 defined by formula_207 
Although formula_208 there do exist ordered vector spaces for which set equality does hold.
Special types of ordered vector spaces.
Let formula_1 be an ordered vector space. We say that an ordered vector space formula_1 is Archimedean ordered and that the order of formula_1 is Archimedean if whenever formula_37 in formula_1 is such that formula_214 is majorized (that is, there exists some formula_215 such that formula_216 for all formula_217) then formula_218 
A topological vector space (TVS) that is an ordered vector space is necessarily Archimedean if its positive cone is closed. 
We say that a preordered vector space formula_1 is regularly ordered and that its order is regular if it is Archimedean ordered and formula_36 distinguishes points in formula_21 
This property guarantees that there are sufficiently many positive linear forms to be able to successfully use the tools of duality to study ordered vector spaces. 
An ordered vector space is called a vector lattice if for all elements formula_37 and formula_223 the supremum formula_224 and infimum formula_225 exist.
Subspaces, quotients, and products.
Throughout let formula_1 be a preordered vector space with positive cone formula_57 
Subspaces
If formula_191 is a vector subspace of formula_1 then the canonical ordering on formula_191 induced by formula_1's positive cone formula_25 is the partial order induced by the pointed convex cone formula_233 where this cone is proper if formula_25 is proper. 
Quotient space
Let formula_191 be a vector subspace of an ordered vector space formula_4 formula_237 be the canonical projection, and let formula_238 
Then formula_239 is a cone in formula_240 that induces a canonical preordering on the quotient space formula_241 
If formula_239 is a proper cone informula_240 then formula_239 makes formula_240 into an ordered vector space. 
If formula_191 is formula_25-saturated then formula_239 defines the canonical order of formula_241 
Note that formula_250 provides an example of an ordered vector space where formula_251 is not a proper cone. 
If formula_1 is also a topological vector space (TVS) and if for each neighborhood formula_253 of the origin in formula_1 there exists a neighborhood formula_255 of the origin such that formula_256 then formula_239 is a normal cone for the quotient topology. 
If formula_1 is a topological vector lattice and formula_191 is a closed solid sublattice of formula_1 then formula_261 is also a topological vector lattice. 
Product
If formula_94 is any set then the space formula_263 of all functions from formula_94 into formula_1 is canonically ordered by the proper cone formula_266 
Suppose that formula_267 is a family of preordered vector spaces and that the positive cone of formula_268 is formula_269 
Then formula_270 is a pointed convex cone in formula_271 which determines a canonical ordering on formula_272 
formula_25 is a proper cone if all formula_274 are proper cones. 
Algebraic direct sum
The algebraic direct sum formula_275 of formula_267 is a vector subspace of formula_277 that is given the canonical subspace ordering inherited from formula_278
If formula_279 are ordered vector subspaces of an ordered vector space formula_1 then formula_1 is the ordered direct sum of these subspaces if the canonical algebraic isomorphism of formula_1 onto formula_283 (with the canonical product order) is an order isomorphism.