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 "X" over the real numbers R and a preorder ≤ on the set "X", the pair is called a preordered vector space and we say that the preorder ≤ is compatible with the vector space structure of "X" and call ≤ a vector preorder on "X" if for all "x", "y", "z" in "X" and in formula_1 the following two axioms are satisfied
If ≤ is a partial order compatible with the vector space structure of "X" then is called an ordered vector space and ≤ is called a vector partial order on "X". 
The two axioms imply that translations and positive homotheties are automorphisms of the order structure and the mapping is an isomorphism to the dual order structure. Ordered vector spaces are ordered groups under their addition operation.
Note that "x" ≤ "y" if and only if −"y" ≤ −"x".
Positive cones and their equivalence to orderings.
A subset "C" of a vector space "X" is called a cone if for all real "r" > 0, "rC" ⊆ "C". 
A cone is called pointed if it contains the origin. 
A cone "C" is convex if and only if "C" + "C" ⊆ "C". 
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 "C" in a vector space "X" is said to be generating if "X" = "C" − "C". 
A positive cone is generating if and only if it is a directed set under ≤. 
Given a preordered vector space "X", the subset "X" + of all elements "x" in ("X", ≤) satisfying "x" ≥ 0 is a pointed convex cone with vertex 0 (i.e. it contains 0) called the positive cone of "X" and denoted by formula_2. 
The elements of the positive cone are called positive. 
If "x" and "y" are elements of a preordered vector space ("X", ≤), then "x" ≤ "y" if and only if "y" − "x" ∈ "X"+. 
Given any pointed convex cone "C" with vertex 0, one may define a preorder ≤ on "X" that is compatible with the vector space structure of "X" by declaring for all "x" and "y" in "X", that "x" ≤ "y" if and only if "y" − "x" ∈ "C"; 
the positive cone of this resulting preordered vector space is "C". 
There is thus a one-to-one correspondence between pointed convex cones with vertex 0 and vector preorders on "X". 
If "X" is preordered then we may form an equivalence relation on "X" by defining "x" is equivalent to "y" if and only if "x" ≤ "y" and "y" ≤ "x"; 
if "N" is the equivalence class containing the origin then "N" is a vector subspace of "X" and "X"/"N" is an ordered vector space under the relation: "A" ≤ "B" if and only there exist "a" in "A" and "b" in "B" such that "a" ≤ "b". 
A subset of "C" of a vector space "X" is called a proper cone if it is a convex cone of vertex 0 satisfying "C" ∩ (−"C") = {0}. 
Explicitly, "C" is a proper cone if (1) "C" + "C" ⊆ "C", (2) "rC" ⊆ "C" for all "r" > 0, and (3) "C" ∩ (−"C") = {0}. 
The intersection of any non-empty family of proper cones is again a proper cone. 
Each proper cone "C" in a real vector space induces an order on the vector space by defining "x" ≤ "y" if and only if "y" − "x" ∈ "C", and furthermore, the positive cone of this ordered vector space will be "C".
Therefore, there exists a one-to-one correspondence between the proper convex cones of "X" and the vector partial orders on "X". 
By a total vector ordering on "X" we mean a total order on "X" that is compatible with the vector space structure of "X". 
The family of total vector orderings on a vector space "X" 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 "R" and "S" are two orderings of a vector space with positive cones "P" and "Q", respectively, then we say that "R" is finer than "S" if "P" ⊆ "Q".
Examples.
The real numbers with the usual ordering form a totally ordered vector space. 
For all integers "n" ≥ 0, the Euclidean space ℝ"n" 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 "n" = 0 or 1. 
Pointwise order.
If "S" is any set and if "X" is a vector space (over the reals) of real-valued functions on "S", then the pointwise order on "X" is given by, for all "f", "g" ∈ "X", "f" ≤ "g" if and only if "f"("s") ≤ "g"("s") for all "s" in "S". 
Spaces that are typically assigned this order include:
The space formula_3 of all measurable almost-everywhere bounded real-valued maps on ℝ, where the preorder is defined for all "f", "g" ∈ formula_3 by "f" ≤ "g" if and only if "f"("s") ≤ "g"("s") almost everywhere.
Intervals and the order bound dual.
An order interval in a preordered vector space is set of the form 
From axioms 1 and 2 above it follows that "x", "y" ∈ ["a", "b"] and 0 < λ < 1 implies λ"x" + (1 − "λ")"y" in ["a", "b"]; 
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 "x" ≥ 0 then the interval of the form [−"x", "x"] is balanced. 
An order unit of a preordered vector space is any element "x" such that the set [−"x", "x"] is absorbing. 
The set of all linear functionals on a preordered vector space "X" that map every order interval into a bounded set is called the order bound dual of "X" and denoted by "X""b". 
If a space is ordered then its order bound dual is a vector subspace of its algebraic dual. 
A subset "A" of an ordered vector space "X" is called order complete if for every non-empty subset "B" ⊆ "A" such that "B" is order bounded in "A", both formula_5 and formula_6 exist and are elements of "A". 
We say that an ordered vector space "X" is order complete is "X" is an order complete subset of "X".
Examples.
If ("X", ≤) is a preordered vector space over the reals with order unit "u", then the map formula_7 is a sublinear functional.
Properties.
If "X" is a preordered vector space then for all "x", "y" ∈ "X", 
Spaces of linear maps.
A cone "C" 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 formula_8 is a proper cone in L("X"; "W"), 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("X"; "W"). 
More generally, if "M" is any vector subspace of L("X"; "W") such that is a proper cone, the ordering defined by is called the canonical ordering of "M". 
Positive functionals and the order dual.
A linear function "f" 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 "C", called the dual cone and denoted by formula_9, is a cone equal to the polar of −"C". 
The preorder induced by the dual cone on the space of linear functionals on "X" is called the dual preorder. 
The order dual of an ordered vector space "X" is the set, denoted by formula_10, defined by formula_11. 
Although formula_12, there do exist ordered vector spaces for which set equality does "not" hold.
Special types of ordered vector spaces.
Let "X" be an ordered vector space. 
We say that an ordered vector space "X" is Archimedean ordered and that the order of "X" is Archimedean if whenever "x" in "X" is such that formula_13 is majorized (i.e. there exists some "y" in "X" such that "nx" ≤ "y" for all formula_14) then . 
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 "X" is regularly ordered and that its order is regular if it is Archimedean ordered and "X" + distinguishes points in "X". 
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 "x" and "y", the supremum sup("x", "y") and infimum inf("x", "y") exist.
Subspaces, quotients, and products.
Throughout let "X" be a preordered vector space with positive cone "C". 
If "M" is a vector subspace of "X" then the canonical ordering on "M" induced by "X"'s positive cone "C" is the partial order induced by the pointed convex cone "C" ∩ "M", where this cone is proper if "C" is proper. 
Let "M" be a vector subspace of an ordered vector space "X", formula_15 be the canonical projection, and let formula_16. 
Then formula_17 is a cone in "X"/"M" that induces a canonical preordering on the quotient space "X"/"M". 
If formula_17 is a proper cone in "X"/"M" then formula_17 makes "X"/"M" into an ordered vector space. 
If "M" is "C"-saturated then formula_17 defines the canonical order of "X"/"M". 
Note that formula_21 provides an example of an ordered vector space where formula_22 is not a proper cone. 
If "X" is also a topological vector space (TVS) and if for each neighborhood "V" of 0 in "X" there exists a neighborhood "U" of 0 such that [("U" + "N") ∩ C] ⊆ "V" + "N" then formula_17 is a normal cone for the quotient topology. 
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. 
If "S" is any set then the space "X""S" of all functions from "S" into "X" is canonically ordered by the proper cone formula_24. 
Suppose that formula_25 is a family of preordered vector spaces and that the positive cone of formula_26 is formula_27. 
Then formula_28 is a pointed convex cone in formula_29, which determines a canonical ordering on formula_29; 
"C" is a proper cone if all formula_27 are proper cones. 
The algebraic direct sum formula_32 of formula_25 is a vector subspace of formula_29 that is given the canonical subspace ordering inherited from formula_29.
If "X"1, ..., "X""n" 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 formula_36 (with the canonical product order) is an order isomorphism.