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{{Short description|Vector space with a partial order}}
[[File:Ordered space illustration.svg|right|thumb|A point <math>x</math> in <math>\Reals^2</math> and the [[Set (mathematics)|set]] of all <math>y</math> such that <math>x \leq y</math> (in red). The order here is <math>x \leq y</math> if and only if <math>x_1 \leq y_1</math> and <math>x_2 \leq y_2.</math>]]
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 <math>X</math> over the [[real number]]s <math>\Reals</math> and a [[preorder]] <math>\,\leq\,</math> on the [[set (mathematics)|set]] <math>X,</math> the pair <math>(X, \leq)</math> is called a '''preordered vector space''' and we say that the preorder <math>\,\leq\,</math> '''is compatible with the vector space structure''' of <math>X</math> and call <math>\,\leq\,</math> a '''vector preorder''' on <math>X</math> if for all <math>x, y, z \in X</math> and <math>r \in \Reals</math> with <math>r \geq 0</math> the following two axioms are satisfied

# <math>x \leq y</math> implies <math>x + z \leq y + z,</math>
# <math>y \leq x</math> implies <math>r y \leq r x.</math>

If <math>\,\leq\,</math> is a [[partial order]] compatible with the vector space structure of <math>X</math> then <math>(X, \leq)</math> is called an '''ordered vector space''' and <math>\,\leq\,</math> is called a '''vector partial order''' on <math>X.</math>
The two axioms imply that [[Translation (geometry)|translation]]s and [[positive homothety|positive homotheties]] are [[automorphism]]s of the order structure and the mapping <math>x \mapsto -x</math> is an [[isomorphism]] to the [[Duality (order theory)|dual order structure]]. Ordered vector spaces are [[ordered group]]s under their addition operation.
Note that <math>x \leq y</math> if and only if <math>-y \leq -x.</math>

==Positive cones and their equivalence to orderings==

A [[subset]] <math>C</math> of a vector space <math>X</math> is called a '''cone''' if for all real <math>r > 0,</math> <math>r C \subseteq C.</math> A cone is called '''pointed''' if it contains the origin. A cone <math>C</math> is convex if and only if <math>C + C \subseteq C.</math> The [[Intersection (set theory)|intersection]] of any [[Empty set|non-empty]] family of cones (resp. convex cones) is again a cone (resp. convex cone);
the same is true of the [[Union (set theory)|union]] of an increasing (under [[subset|set inclusion]]) family of cones (resp. convex cones). A cone <math>C</math> in a vector space <math>X</math> is said to be '''generating''' if <math>X = C - C.</math>{{sfn|Schaefer|Wolff|1999|pp=250-257}}
A positive cone is generating if and only if it is a [[directed set]] under <math>\,\leq.</math>

Given a preordered vector space <math>X,</math> the subset <math>X^+</math> of all elements <math>x</math> in <math>(X, \leq)</math> satisfying <math>x \geq 0</math> is a pointed [[convex cone]] with vertex <math>0</math> (that is, it contains <math>0</math>) called the '''positive cone''' of <math>X</math> and denoted by <math>\operatorname{PosCone} X.</math>
The elements of the positive cone are called '''positive'''.
If <math>x</math> and <math>y</math> are elements of a preordered vector space <math>(X, \leq),</math> then <math>x \leq y</math> if and only if <math>y - x \in X^+.</math>
Given any pointed convex cone <math>C</math> with vertex <math>0,</math> one may define a preorder <math>\,\leq\,</math> on <math>X</math> that is compatible with the vector space structure of <math>X</math> by declaring for all <math>x, y \in X,</math> that <math>x \leq y</math> if and only if <math>y - x \in C;</math>
the positive cone of this resulting preordered vector space is <math>C.</math>
There is thus a one-to-one correspondence between pointed convex cones with vertex <math>0</math> and vector preorders on <math>X.</math>{{sfn|Schaefer|Wolff|1999|pp=250-257}}
If <math>X</math> is preordered then we may form an [[equivalence relation]] on <math>X</math> by defining <math>x</math> is equivalent to <math>y</math> if and only if <math>x \leq y</math> and <math>y \leq x;</math>
if <math>N</math> is the [[equivalence class]] containing the origin then <math>N</math> is a vector subspace of <math>X</math> and <math>X / N</math> is an ordered vector space under the relation: <math>A \leq B</math> if and only there exist <math>a \in A</math> and <math>b \in B</math> such that <math>a \leq b.</math>{{sfn|Schaefer|Wolff|1999|pp=250-257}}

A subset of <math>C</math> of a vector space <math>X</math> is called a '''[[proper cone]]''' if it is a convex cone of vertex <math>0</math> satisfying <math>C \cap (- C) = \{0\}.</math>
Explicitly, <math>C</math> is a proper cone if (1) <math>C + C \subseteq C,</math> (2) <math>r C \subseteq C</math> for all <math>r > 0,</math> and (3) <math>C \cap (- C) = \{0\}.</math>{{sfn|Schaefer|Wolff|1999|pp=205–209}}
The intersection of any non-empty family of proper cones is again a proper cone. Each proper cone <math>C</math> in a real vector space induces an order on the vector space by defining <math>x \leq y</math> if and only if <math>y - x \in C,</math> and furthermore, the positive cone of this ordered vector space will be <math>C.</math> Therefore, there exists a one-to-one correspondence between the proper convex cones of <math>X</math> and the vector partial orders on <math>X.</math>

By a '''total vector ordering''' on <math>X</math> we mean a [[total order]] on <math>X</math> that is compatible with the vector space structure of <math>X.</math>
The family of total vector orderings on a vector space <math>X</math> is in one-to-one correspondence with the family of all proper cones that are maximal under set inclusion.{{sfn|Schaefer|Wolff|1999|pp=250-257}}
A total vector ordering ''cannot'' be [[Archimedean order|Archimedean]] if its [[dimension (vector space)|dimension]], when considered as a vector space over the reals, is greater than 1.{{sfn|Schaefer|Wolff|1999|pp=250-257}}

If <math>R</math> and <math>S</math> are two orderings of a vector space with positive cones <math>P</math> and <math>Q,</math> respectively, then we say that <math>R</math> is '''finer''' than <math>S</math> if <math>P \subseteq Q.</math>{{sfn|Schaefer|Wolff|1999|pp=205–209}}

==Examples==

The real numbers with the usual ordering form a totally ordered vector space. For all [[integer]]s <math>n \geq 0,</math> the [[Euclidean space]] <math>\Reals^n</math> considered as a vector space over the reals with the [[lexicographic order]]ing forms a preordered vector space whose order is [[Archimedean ordered vector space|Archimedean]] if and only if <math>n = 1</math>.{{sfn|Narici|Beckenstein|2011|pp=139-153}}

===Pointwise order===

If <math>S</math> is any set and if <math>X</math> is a vector space (over the reals) of real-valued [[function (mathematics)|functions]] on <math>S,</math> then the '''pointwise order''' on <math>X</math> is given by, for all <math>f, g \in X,</math> <math>f \leq g</math> if and only if <math>f(s) \leq g(s)</math> for all <math>s \in S.</math>{{sfn|Narici|Beckenstein|2011|pp=139-153}}

Spaces that are typically assigned this order include:
* the space <math>\ell^\infty(S, \Reals)</math> of [[bounded function|bounded]] real-valued maps on <math>S.</math>
* the space <math>c_0(\Reals)</math> of real-valued [[sequences]] that [[Limit of a sequence|converge]] to <math>0.</math>
* the space <math>C(S, \Reals)</math> of [[Continuous function (topology)|continuous]] real-valued functions on a [[topological space]] <math>S.</math>
* for any non-negative integer <math>n,</math> the Euclidean space <math>\Reals^n</math> when considered as the space <math>C(\{1, \dots, n\}, \Reals)</math> where <math>S = \{1, \dots, n\}</math> is given the [[discrete topology]].

The space <math>\mathcal{L}^\infty(\Reals, \Reals)</math> of all [[measurable function|measurable]] [[almost-everywhere]] bounded real-valued maps on <math>\Reals,</math> where the preorder is defined for all <math>f, g \in \mathcal{L}^\infty(\Reals, \Reals)</math> by <math>f \leq g</math> if and only if <math>f(s) \leq g(s)</math> almost everywhere.{{sfn|Narici|Beckenstein|2011|pp=139-153}}

==Intervals and the order bound dual==

An '''order interval''' in a preordered vector space is set of the form
<math display=block>\begin{alignat}{4}
[a, b] &= \{x : a \leq x \leq b\}, \\[0.1ex]
[a, b[ &= \{x : a \leq x < b\}, \\
]a, b] &= \{x : a < x \leq b\}, \text{ or } \\
]a, b[ &= \{x : a < x < b\}. \\
\end{alignat}</math>
From axioms 1 and 2 above it follows that <math>x, y \in [a, b]</math> and <math>0 < t < 1</math> implies <math>t x + (1 - t) y</math> belongs to <math>[a, b];</math>
thus these order intervals are convex.
A subset is said to be '''order bounded''' if it is contained in some order interval.{{sfn|Schaefer|Wolff|1999|pp=205–209}}
In a preordered real vector space, if for <math>x \geq 0</math> then the interval of the form <math>[-x, x]</math> is [[balanced set|balanced]].{{sfn|Schaefer|Wolff|1999|pp=205–209}}
An '''[[order unit]]''' of a preordered vector space is any element <math>x</math> such that the set <math>[-x, x]</math> is [[Absorbing set|absorbing]].{{sfn|Schaefer|Wolff|1999|pp=205–209}}

The set of all [[linear functional]]s on a preordered vector space <math>X</math> that map every order interval into a bounded set is called the '''[[order bound dual]]''' of <math>X</math> and denoted by <math>X^{\operatorname{b}}.</math>{{sfn|Schaefer|Wolff|1999|pp=205–209}}
If a space is ordered then its order bound dual is a vector subspace of its [[algebraic dual]].

A subset <math>A</math> of an ordered vector space <math>X</math> is called '''[[order complete]]''' if for every non-empty subset <math>B \subseteq A</math> such that <math>B</math> is order bounded in <math>A,</math> both <math>\sup B</math> and <math>\inf B</math> exist and are elements of <math>A.</math> We say that an ordered vector space <math>X</math> is '''order complete''' is <math>X</math> is an order complete subset of <math>X.</math>{{sfn|Schaefer|Wolff|1999|pp=204-214}}

===Examples===

If <math>(X, \leq)</math> is a preordered vector space over the reals with order unit <math>u,</math> then the map <math>p(x) := \inf \{t \in \Reals : x \leq t u\}</math> is a [[sublinear functional]].{{sfn|Narici|Beckenstein|2011|pp=139-153}}

==Properties==

If <math>X</math> is a preordered vector space then for all <math>x, y \in X,</math>

* <math>x \geq 0</math> and <math>y \geq 0</math> imply <math>x + y \geq 0.</math>{{sfn|Narici|Beckenstein|2011|pp=139-153}}
* <math>x \leq y</math> if and only if <math>-y \leq -x.</math>{{sfn|Narici|Beckenstein|2011|pp=139-153}}
* <math>x \leq y</math> and <math>r < 0</math> imply <math>r x \geq r y.</math>{{sfn|Narici|Beckenstein|2011|pp=139-153}}
* <math>x \leq y</math> if and only if <math>y = \sup \{x, y\}</math> if and only if <math>x = \inf \{x, y\}</math>{{sfn|Narici|Beckenstein|2011|pp=139-153}}
* <math>\sup \{x, y\}</math> exists if and only if <math>\inf \{-x, -y\}</math> exists, in which case <math>\inf \{-x, -y\} = - \sup \{x, y\}.</math>{{sfn|Narici|Beckenstein|2011|pp=139-153}}
* <math>\sup \{x, y\}</math> exists if and only if <math>\inf \{x, y\}</math> exists, in which case for all <math>z \in X,</math>{{sfn|Narici|Beckenstein|2011|pp=139-153}}
** <math>\sup \{x + z, y + z\} = z + \sup \{x, y\},</math> and
** <math>\inf \{x + z, y + z\} = z + \inf \{x, y\}</math>
** <math>x + y = \inf\{x, y\} + \sup \{x, y\}.</math>
* <math>X</math> is a [[vector lattice]] if and only if <math>\sup \{0, x\}</math> exists for all <math>x \in X.</math>{{sfn|Narici|Beckenstein|2011|pp=139-153}}

==Spaces of linear maps==
{{Main|Positive linear operator}}

A cone <math>C</math> is said to be '''generating''' if <math>C - C</math> is equal to the whole vector space.{{sfn|Schaefer|Wolff|1999|pp=205–209}}
If <math>X</math> and <math>W</math> are two non-trivial ordered vector spaces with respective positive cones <math>P</math> and <math>Q,</math> then <math>P</math> is generating in <math>X</math> if and only if the set <math>C = \{u \in L(X; W) : u(P) \subseteq Q\}</math> is a proper cone in <math>L(X; W),</math> which is the space of all linear maps from <math>X</math> into <math>W.</math>
In this case, the ordering defined by <math>C</math> is called the '''canonical ordering''' of <math>L(X; W).</math>{{sfn|Schaefer|Wolff|1999|pp=205–209}}
More generally, if <math>M</math> is any vector subspace of <math>L(X; W)</math> such that <math>C \cap M</math> is a proper cone, the ordering defined by <math>C \cap M</math> is called the '''canonical ordering''' of <math>M.</math>{{sfn|Schaefer|Wolff|1999|pp=205–209}}

===Positive functionals and the order dual===

A [[linear function]] <math>f</math> on a preordered vector space is called '''positive''' if it satisfies either of the following equivalent conditions:

# <math>x \geq 0</math> implies <math>f(x) \geq 0.</math>
# if <math>x \leq y</math> then <math>f(x) \leq f(y).</math>{{sfn|Narici|Beckenstein|2011|pp=139-153}}

The set of all positive linear forms on a vector space with positive cone <math>C,</math> called the '''[[Dual cone and polar cone|dual cone]]''' and denoted by <math>C^*,</math> is a cone equal to the [[Polar set|polar]] of <math>-C.</math>
The preorder induced by the dual cone on the space of linear functionals on <math>X</math> is called the '''{{visible anchor|dual preorder}}'''.{{sfn|Narici|Beckenstein|2011|pp=139-153}}

The '''[[Order dual (functional analysis)|order dual]]''' of an ordered vector space <math>X</math> is the set, denoted by <math>X^+,</math> defined by <math>X^+ := C^* - C^*.</math>
Although <math>X^+ \subseteq X^b,</math> there do exist ordered vector spaces for which set equality does {{em|not}} hold.{{sfn|Schaefer|Wolff|1999|pp=205–209}}

==Special types of ordered vector spaces==

Let <math>X</math> be an ordered vector space. We say that an ordered vector space <math>X</math> is '''[[Archimedean ordered vector space|Archimedean ordered]]''' and that the order of <math>X</math> is '''Archimedean''' if whenever <math>x</math> in <math>X</math> is such that <math>\{n x : n \in \N\}</math> is '''[[Majorization|majorized]]''' (that is, there exists some <math>y \in X</math> such that <math>n x \leq y</math> for all <math>n \in \N</math>) then <math>x \leq 0.</math>{{sfn|Schaefer|Wolff|1999|pp=205–209}}
A [[topological vector space]] (TVS) that is an ordered vector space is necessarily Archimedean if its positive cone is closed.{{sfn|Schaefer|Wolff|1999|pp=205–209}}

We say that a preordered vector space <math>X</math> is '''[[regularly ordered]]''' and that its order is '''regular''' if it is [[Archimedean ordered]] and <math>X^+</math> distinguishes points in <math>X.</math>{{sfn|Schaefer|Wolff|1999|pp=205–209}}
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.{{sfn|Schaefer|Wolff|1999|pp=205–209}}

An ordered vector space is called a '''[[vector lattice]]''' if for all elements <math>x</math> and <math>y,</math> the [[supremum]] <math>\sup (x, y)</math> and [[infimum]] <math>\inf (x, y)</math> exist.{{sfn|Schaefer|Wolff|1999|pp=205–209}}

==Subspaces, quotients, and products==

Throughout let <math>X</math> be a preordered vector space with positive cone <math>C.</math>

'''Subspaces'''

If <math>M</math> is a vector subspace of <math>X</math> then the canonical ordering on <math>M</math> induced by <math>X</math>'s positive cone <math>C</math> is the partial order induced by the pointed convex cone <math>C \cap M,</math> where this cone is proper if <math>C</math> is proper.{{sfn|Schaefer|Wolff|1999|pp=205–209}}

'''Quotient space'''

Let <math>M</math> be a vector subspace of an ordered vector space <math>X,</math> <math>\pi : X \to X / M</math> be the canonical projection, and let <math>\hat{C} := \pi(C).</math>
Then <math>\hat{C}</math> is a cone in <math>X / M</math> that induces a canonical preordering on the [[quotient space (linear algebra)|quotient space]] <math>X / M.</math>
If <math>\hat{C}</math> is a proper cone in<math>X / M</math> then <math>\hat{C}</math> makes <math>X / M</math> into an ordered vector space.{{sfn|Schaefer|Wolff|1999|pp=205–209}}
If <math>M</math> is [[Cone-saturated|<math>C</math>-saturated]] then <math>\hat{C}</math> defines the canonical order of <math>X / M.</math>{{sfn|Schaefer|Wolff|1999|pp=250-257}}
Note that <math>X = \Reals^2_0</math> provides an example of an ordered vector space where <math>\pi(C)</math> is not a proper cone.

If <math>X</math> is also a [[topological vector space]] (TVS) and if for each [[neighborhood (mathematics)|neighborhood]] <math>V</math> of the origin in <math>X</math> there exists a neighborhood <math>U</math> of the origin such that <math>[(U + N) \cap C] \subseteq V + N</math> then <math>\hat{C}</math> is a [[Normal cone (functional analysis)|normal cone]] for the [[quotient topology]].{{sfn|Schaefer|Wolff|1999|pp=250-257}}

If <math>X</math> is a [[topological vector lattice]] and <math>M</math> is a closed [[Solid set|solid]] sublattice of <math>X</math> then <math>X / L</math> is also a topological vector lattice.{{sfn|Schaefer|Wolff|1999|pp=250-257}}

'''Product'''

If <math>S</math> is any set then the space <math>X^S</math> of all functions from <math>S</math> into <math>X</math> is canonically ordered by the proper cone <math>\left\{f \in X^S : f(s) \in C \text{ for all } s \in S\right\}.</math>{{sfn|Schaefer|Wolff|1999|pp=205–209}}

Suppose that <math>\left\{X_\alpha : \alpha \in A\right\}</math> is a family of preordered vector spaces and that the positive cone of <math>X_\alpha</math> is <math>C_\alpha.</math>
Then <math display="inline">C := \prod_\alpha C_\alpha</math> is a pointed convex cone in <math display="inline">\prod_\alpha X_\alpha,</math> which determines a canonical ordering on <math display="inline">\prod_\alpha X_\alpha;</math>
<math>C</math> is a proper cone if all <math>C_\alpha</math> are proper cones.{{sfn|Schaefer|Wolff|1999|pp=205–209}}

'''Algebraic direct sum'''

The algebraic [[direct sum]] <math display="inline">\bigoplus_\alpha X_\alpha</math> of <math>\left\{X_\alpha : \alpha \in A\right\}</math> is a vector subspace of <math display="inline">\prod_\alpha X_\alpha</math> that is given the canonical subspace ordering inherited from <math display="inline">\prod_\alpha X_\alpha.</math>{{sfn|Schaefer|Wolff|1999|pp=205–209}}
If <math>X_1, \dots, X_n</math> are ordered vector subspaces of an ordered vector space <math>X</math> then <math>X</math> is the ordered direct sum of these subspaces if the canonical algebraic isomorphism of <math>X</math> onto <math>\prod_\alpha X_\alpha</math> (with the canonical product order) is an [[order isomorphism]].{{sfn|Schaefer|Wolff|1999|pp=205–209}}

==Examples==

* The [[real number]]s with the usual order is an ordered vector space.
* <math>\Reals^2</math> is an ordered vector space with the <math>\,\leq\,</math> relation defined in any of the following ways (in order of increasing strength, that is, decreasing sets of pairs):
** [[Lexicographical order]]: <math>(a, b) \leq (c, d)</math> if and only if <math>a < c</math> or <math>(a = c \text{ and } b \leq d).</math> This is a [[total order]]. The positive cone is given by <math>x > 0</math> or <math>(x = 0 \text{ and } y \leq 0),</math> that is, in [[Polar coordinate system|polar coordinates]], the set of points with the angular coordinate satisfying <math>-\pi / 2 < \theta \leq \pi / 2,</math> together with the origin.
** <math>(a, b) \leq (c, d)</math> if and only if <math>a \leq c</math> and <math>b \leq d</math> (the [[product order]] of two copies of <math>\Reals</math> with <math>\leq</math>). This is a partial order. The positive cone is given by <math>x \geq 0</math> and <math>y \geq 0,</math> that is, in polar coordinates <math>0 \leq \theta \leq \pi / 2,</math> together with the origin.
** <math>(a, b) \leq (c, d)</math> if and only if <math>(a < c \text{ and } b < d)</math> or <math>(a = c \text{ and } b = d)</math> (the [[reflexive closure]] of the [[Direct product#Direct product of binary relations|direct product]] of two copies of <math>\Reals</math> with "<"). This is also a partial order. The positive cone is given by <math>(x > 0 \text{ and } y > 0)</math> or <math>x = y = 0),</math> that is, in polar coordinates, <math>0 < \theta < \pi / 2,</math> together with the origin.
:Only the second order is, as a subset of <math>\Reals^4,</math> closed; see [[Partially ordered set#Partial orders in topological spaces|partial orders in topological spaces]].
:For the third order the two-dimensional "[[Partially ordered set#Intervals|intervals]]" <math>p < x < q</math> are [[open set]]s which generate the topology.
* <math>\Reals^n</math> is an ordered vector space with the <math>\,\leq\,</math> relation defined similarly. For example, for the second order mentioned above:
** <math>x \leq y</math> if and only if <math>x_i \leq y_i</math> for <math>i = 1, \dots, n.</math>
* A [[Riesz space]] is an ordered vector space where the order gives rise to a [[lattice (order)|lattice]].
* The space of continuous functions on <math>[0, 1]</math> where <math>f \leq g</math> if and only if <math>f(x) \leq g(x)</math> for all <math>x</math> in <math>[0, 1].</math>

==See also==

* {{annotated link|Order topology (functional analysis)}}
* {{annotated link|Ordered field}}
* {{annotated link|Ordered group}}
* {{annotated link|Ordered ring}}
* {{annotated link|Ordered topological vector space}}
* {{annotated link|Partially ordered space}}
* {{annotated link|Product order}}
* {{annotated link|Riesz space}}
* {{annotated link|Topological vector lattice}}
* {{annotated link|Vector lattice}}

==References==

{{reflist}}

==Bibliography==

* {{cite book|last=Aliprantis|first=Charalambos D|authorlink=Charalambos D. Aliprantis|author2=Burkinshaw, Owen|title=Locally solid Riesz spaces with applications to economics|edition=Second|publisher=Providence, R. I.: American Mathematical Society|year=2003|pages=|isbn=0-8218-3408-8}}
* [[Bourbaki, Nicolas]]; <cite>Elements of Mathematics: Topological Vector Spaces</cite>; {{isbn|0-387-13627-4}}.
* {{Narici Beckenstein Topological Vector Spaces|edition=2}} 
* {{Schaefer Wolff Topological Vector Spaces|edition=2}} 
* {{cite book|author=Wong|title=Schwartz spaces, nuclear spaces, and tensor products|publisher=Springer-Verlag|publication-place=Berlin New York|year=1979|isbn=3-540-09513-6|oclc=5126158}} 

{{Order theory}}
{{Ordered topological vector spaces}}
{{Functional analysis}}

[[Category:Functional analysis]]
[[Category:Ordered groups]]
[[Category:Vector spaces]]

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