A point in and the set of all such that (in red). The order here is if and only if and

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 over the real numbers and a preorder on the set the pair is called a preordered vector space and we say that the preorder is compatible with the vector space structure of and call a vector preorder on if for all and with the following two axioms are satisfied

  1. implies
  2. implies

If is a partial order compatible with the vector space structure of then is called an ordered vector space and is called a vector partial order on 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 if and only if

Positive cones and their equivalence to orderings

A subset of a vector space is called a cone if for all real A cone is called pointed if it contains the origin. A cone is convex if and only if 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 in a vector space is said to be generating if [1]

Given a preordered vector space the subset of all elements in satisfying is a pointed convex cone with vertex (that is, it contains ) called the positive cone of and denoted by The elements of the positive cone are called positive. If and are elements of a preordered vector space then if and only if The positive cone is generating if and only if is a directed set under Given any pointed convex cone with vertex one may define a preorder on that is compatible with the vector space structure of by declaring for all that if and only if the positive cone of this resulting preordered vector space is There is thus a one-to-one correspondence between pointed convex cones with vertex and vector preorders on [1] If is preordered then we may form an equivalence relation on by defining is equivalent to if and only if and if is the equivalence class containing the origin then is a vector subspace of and is an ordered vector space under the relation: if and only there exist and such that [1]

A subset of of a vector space is called a proper cone if it is a convex cone of vertex satisfying Explicitly, is a proper cone if (1) (2) for all and (3) [2] The intersection of any non-empty family of proper cones is again a proper cone. Each proper cone in a real vector space induces an order on the vector space by defining if and only if and furthermore, the positive cone of this ordered vector space will be Therefore, there exists a one-to-one correspondence between the proper convex cones of and the vector partial orders on

By a total vector ordering on we mean a total order on that is compatible with the vector space structure of The family of total vector orderings on a vector space is in one-to-one correspondence with the family of all proper cones that are maximal under set inclusion.[1] A total vector ordering cannot be Archimedean if its dimension, when considered as a vector space over the reals, is greater than 1.[1]

If and are two orderings of a vector space with positive cones and respectively, then we say that is finer than if [2]

Examples

The real numbers with the usual ordering form a totally ordered vector space. For all integers the Euclidean space 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 .[3]

Pointwise order

If is any set and if is a vector space (over the reals) of real-valued functions on then the pointwise order on is given by, for all if and only if for all [3]

Spaces that are typically assigned this order include:

The space of all measurable almost-everywhere bounded real-valued maps on where the preorder is defined for all by if and only if almost everywhere.[3]

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 and implies belongs to thus these order intervals are convex. A subset is said to be order bounded if it is contained in some order interval.[2] In a preordered real vector space, if for then the interval of the form is balanced.[2] An order unit of a preordered vector space is any element such that the set is absorbing.[2]

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 [2] If a space is ordered then its order bound dual is a vector subspace of its algebraic dual.

A subset of an ordered vector space 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 an ordered vector space is order complete is is an order complete subset of [4]

Examples

If is a preordered vector space over the reals with order unit then the map is a sublinear functional.[3]

Properties

If is a preordered vector space then for all

Spaces of linear maps

Main article: Positive linear operator

A cone is said to be generating if is equal to the whole vector space.[2] 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 [2] More generally, if is any vector subspace of such that is a proper cone, the ordering defined by is called the canonical ordering of [2]

Positive functionals and the order dual

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

  1. implies
  2. if then [3]

The set of all positive linear forms on a vector space with positive cone called the dual cone and denoted by is a cone equal to the polar of The preorder induced by the dual cone on the space of linear functionals on is called the dual preorder.[3]

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.[2]

Special types of ordered vector spaces

Let be an ordered vector space. We say that an ordered vector space is Archimedean ordered and that the order of is Archimedean if whenever in is such that is majorized (that is, there exists some such that for all ) then [2] A topological vector space (TVS) that is an ordered vector space is necessarily Archimedean if its positive cone is closed.[2]

We say that a preordered vector space is regularly ordered and that its order is regular if it is Archimedean ordered and distinguishes points in [2] 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.[2]

An ordered vector space is called a vector lattice if for all elements and the supremum and infimum exist.[2]

Subspaces, quotients, and products

Throughout let be a preordered vector space with positive cone

Subspaces

If is a vector subspace of then the canonical ordering on induced by 's positive cone is the partial order induced by the pointed convex cone where this cone is proper if is proper.[2]

Quotient space

Let be a vector subspace of an ordered vector space 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.[2] If is -saturated then defines the canonical order of [1] Note that provides an example of an ordered vector space where is not a proper cone.

If is also a topological vector space (TVS) and if for each neighborhood of the origin in there exists a neighborhood of the origin such that then is a normal cone for the quotient topology.[1]

If is a topological vector lattice and is a closed solid sublattice of then is also a topological vector lattice.[1]

Product

If is any set then the space of all functions from into is canonically ordered by the proper cone [2]

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.[2]

Algebraic direct sum

The algebraic direct sum of is a vector subspace of that is given the canonical subspace ordering inherited from [2] 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.[2]

Examples

Only the second order is, as a subset of closed; see partial orders in topological spaces.
For the third order the two-dimensional "intervals" are open sets which generate the topology.

See also

References

  1. ^ a b c d e f g h Schaefer & Wolff 1999, pp. 250–257.
  2. ^ a b c d e f g h i j k l m n o p q r s t u Schaefer & Wolff 1999, pp. 205–209.
  3. ^ a b c d e f g h i j k l m Narici & Beckenstein 2011, pp. 139–153.
  4. ^ Schaefer & Wolff 1999, pp. 204–214.

Bibliography