In mathematics, an ordered vector space or partially ordered vector space is a real vector space equipped with a partial order that is compatible with the vector space operations.

Definition

over the real numbers and a preorder on the set is called a preordered vector space is compatible with the vector space structure a vector preorder the following two axioms are satisfied

is a partial order is called an ordered vector space is called a vector partial order The two axioms imply that translationss and positive homotheties are automorphisms is an isomorphism to the dual order structure. Ordered vector spaces are ordered groups

Positive cones and their equivalence to orderings

A subset is called a cone. A cone is called pointed 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 is said to be generating

is a pointed convex cone) called the positive cone The elements of the positive cone are called positive is a directed set is preordered then we may form an equivalence relation is the equivalence class

is called a proper cone

By a total vector ordering we mean a total order 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.

is finer

Intervals and the order bound dual

An order interval thus these order intervals are convex. A subset is said to be order bounded if it is contained in some order interval. is balanced. An order unit is absorbing.

The set of all linear functionals that map every order interval into a bounded set is called the order bound dual If a space is ordered then its order bound dual is a vector subspace of its algebraic dual.

is called order complete is order complete

Examples

is a sublinear functional.

Properties

• and

• is a vector lattice

Spaces of linear maps

is said to be generating is equal to the whole vector space. is called the canonical ordering is called the canonical ordering

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:

called the dual cone is a cone equal to the polar is called the .

The order dual there do exist ordered vector spaces for which set equality does hold.

Special types of ordered vector spaces

is Archimedean ordered is Archimedean is majorized A topological vector space (TVS) that is an ordered vector space is necessarily Archimedean if its positive cone is closed.

is regularly ordered and that its order is regular if it is Archimedean ordered 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 the supremum and infimum exist.

Subspaces, quotients, and products

Subspaces

is proper.

Quotient space

that induces a canonical preordering on the quotient space into an ordered vector space. is C-saturated is not a proper cone.

is also a topological vector space (TVS) and if for each neighborhood is a normal cone for the quotient topology.

is a topological vector lattice is a closed solid is also a topological vector lattice.

Product

are proper cones.

Algebraic direct sum

The algebraic direct sum (with the canonical product order) is an order isomorphism.

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.

relation defined in any of the following ways (in order of increasing strength, that is, decreasing sets of pairs):

• Lexicographical order This is a total order that is, in polar coordinates together with the origin.

• (the product order together with the origin.

• (the reflexive closure of the direct product together with the origin.

• closed; see partial orders in topological spaces.

• For the third order the two-dimensional "intervals are open sets which generate the topology.

relation defined similarly. For example, for the second order mentioned above:

• A Riesz space is an ordered vector space where the order gives rise to a lattice.

• matrices with real entries. The Loewner order is positive semi-definite. Its positive cone is, by definition, the set of all positive definite matrices. Furthermore, the spectral theorem applied to symmetric matrices establishes that this cone is generating.

Pointwise order

is a vector space (over the reals) of real-valued functions then the pointwise order

Spaces that are typically assigned this order include:

• of bounded

• of real-valued sequences that converge

• of continuous real-valued functions on a topological space

• is given the discrete topology.

of all measurable almost-everywhere almost everywhere.

See also

References

Bibliography

• Bourbaki, Nicolas; Elements of Mathematics: Topological Vector Spaces; .