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 the following two axioms are satisfied
  1. implies
  2. implies.
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 is again a cone ;
the same is true of the union of an increasing family of 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 ' satisfying x ≥ 0 is a pointed convex cone with vertex 0 called the positive cone of X and denoted by.
The elements of the positive cone are called positive.
If x and y are elements of a preordered vector space, 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 ∩ =.
Explicitly, C is a proper cone if C + C ⊆ C, rC ⊆ C for all r > 0, and C ∩ =.
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 ℝn 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 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 ≤ g for all s ∈ S.
Spaces that are typically assigned this order include:
The space of all measurable almost-everywhere bounded real-valued maps on ℝ, where the preorder is defined for all f, g ∈ by f ≤ g if and only if f ≤ g 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 ∈ and 0 < λ < 1 implies λx + y in ;
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 is balanced.
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 X that map every order interval into a bounded set is called the order bound dual of X and denoted by Xb
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 and 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 is a preordered vector space over the reals with order unit u, then the map 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 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.

Positive functionals and the order dual

A linear function f on a preordered vector space is called positive if it satisfies any of the following equivalent conditions:

  1. x ≥ 0 implies f ≥ 0.
  2. if x ≤ y then f ≤ f.
The set of all positive linear forms on a vector space with positive cone C, called the dual cone and denoted by, 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, defined by.
Although, 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 is majorized then x ≤ 0.
A topological vector space 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 and infimum inf exist.

Subspaces, quotients, and products

Throughout let X be a preordered vector space with positive cone C.
;Subspaces
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.
;Quotient space
Let M be a vector subspace of an ordered vector space X, 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 also a topological vector space and if for each neighborhood V of 0 in X there exists a neighborhood U of 0 such that ⊆ V + N then 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.
;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.

Examples