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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.

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

- implies
- 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

A subset of a vector space is called a **cone** if for all real that is, for all we have . 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 (that is, a convex cone containing ) 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 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 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 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]}

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]}

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 bounded real-valued maps on
- the space of real-valued sequences that converge to
- the space of continuous real-valued functions on a topological space
- for any non-negative integer the Euclidean space when considered as the space where is given the discrete topology.

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]}

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]}

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

If is a preordered vector space then for all

- and imply
^{[3]} - if and only if
^{[3]} - and imply
^{[3]} - if and only if if and only if
^{[3]} - exists if and only if exists, in which case
^{[3]} - exists if and only if exists, in which case for all
^{[3]}- and

- is a vector lattice if and only if exists for all
^{[3]}

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]}

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

- implies
- 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]}

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]}

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]}

- The real numbers with the usual order is an ordered vector space.
- is an ordered vector space with the relation defined in any of the following ways (in order of increasing strength, that is, decreasing sets of pairs):
- Lexicographical order: if and only if or This is a total order. The positive cone is given by or that is, in polar coordinates, the set of points with the angular coordinate satisfying together with the origin.
- if and only if and (the product order of two copies of with ). This is a partial order. The positive cone is given by and that is, in polar coordinates together with the origin.
- if and only if or (the reflexive closure of the direct product of two copies of with "<"). This is also a partial order. The positive cone is given by or that is, in polar coordinates, together with the origin.

- 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.

- is an ordered vector space with the relation defined similarly. For example, for the second order mentioned above:
- if and only if for

- A Riesz space is an ordered vector space where the order gives rise to a lattice.
- The space of continuous functions on where if and only if for all in

- Order topology (functional analysis) – Topology of an ordered vector space
- Ordered field – Algebraic object with an ordered structure
- Ordered group – Group with a compatible partial order
- Ordered ring – ring with a compatible total order
- Ordered topological vector space
- Partially ordered space – Partially ordered topological space
- Product order
- Riesz space – Partially ordered vector space, ordered as a lattice
- Topological vector lattice
- Vector lattice – Partially ordered vector space, ordered as a lattice

- ^
^{a}^{b}^{c}^{d}^{e}^{f}^{g}^{h}Schaefer & Wolff 1999, pp. 250–257. - ^
^{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. - ^
^{a}^{b}^{c}^{d}^{e}^{f}^{g}^{h}^{i}^{j}^{k}^{l}^{m}Narici & Beckenstein 2011, pp. 139–153. **^**Schaefer & Wolff 1999, pp. 204–214.

- Aliprantis, Charalambos D; Burkinshaw, Owen (2003).
*Locally solid Riesz spaces with applications to economics*(Second ed.). Providence, R. I.: American Mathematical Society. ISBN 0-8218-3408-8. - Bourbaki, Nicolas; Elements of Mathematics: Topological Vector Spaces; ISBN 0-387-13627-4.
- Narici, Lawrence; Beckenstein, Edward (2011).
*Topological Vector Spaces*. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. - Schaefer, Helmut H.; Wolff, Manfred P. (1999).
*Topological Vector Spaces*. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135. - Wong (1979).
*Schwartz spaces, nuclear spaces, and tensor products*. Berlin New York: Springer-Verlag. ISBN 3-540-09513-6. OCLC 5126158.