Ordered vector space

A point $x$ in ${ℝ}^2$ and the set of all $y$ such that $x\le y$ (in red). The order here is $x\le y$ if and only if $x_1\le y_1$ and $x_2\le y_2.$

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

Given a vector space $X$ over the real numbers $ℝ$ and a preorder $\le$ on the set $X,$ the pair $(X,\le )$ is called a preordered vector space and we say that the preorder $\le$ is compatible with the vector space structure of $X$ and call $\le$ a vector preorder on $X$ if for all $x,y,z\in X$ and $r\in ℝ$ with $r\ge 0$ the following two axioms are satisfied

1. $x\le y$ implies $x+z\le y+z,$
2. $y\le x$ implies $ry\le rx.$

If $\le$ is a partial order compatible with the vector space structure of $X$ then $(X,\le )$ is called an ordered vector space and $\le$ 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 $x\mapsto -x$ is an isomorphism to the dual order structure. Ordered vector spaces are ordered groups under their addition operation. Note that $x\le y$ if and only if $-y\le -x.$

Positive cones and their equivalence to orderings

A subset $C$ of a vector space $X$ is called a cone if for all real $r>0,$ $rC\subseteq C$. A cone is called pointed if it contains the origin. A cone $C$ is convex if and only if $C+C\subseteq C.$ 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 $C$ in a vector space $X$ is said to be generating if $X=C-C.$Template:Sfn

Given a preordered vector space $X,$ the subset $X^{+}$ of all elements $x$ in $(X,\le )$ satisfying $x\ge 0$ is a pointed convex cone (that is, a convex cone containing $0$) called the positive cone of $X$ and denoted by $\mathrm{PosCone}X.$ The elements of the positive cone are called positive. If $x$ and $y$ are elements of a preordered vector space $(X,\le ),$ then $x\le y$ if and only if $y-x\in X^{+}.$ The positive cone is generating if and only if $X$ is a directed set under $\le .$ Given any pointed convex cone $C$ one may define a preorder $\le$ on $X$ that is compatible with the vector space structure of $X$ by declaring for all $x,y\in X,$ that $x\le y$ if and only if $y-x\in C;$ the positive cone of this resulting preordered vector space is $C.$ There is thus a one-to-one correspondence between pointed convex cones and vector preorders on $X.$Template:Sfn If $X$ is preordered then we may form an equivalence relation on $X$ by defining $x$ is equivalent to $y$ if and only if $x\le y$ and $y\le x;$ 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: $A\le B$ if and only there exist $a\in A$ and $b\in B$ such that $a\le b.$Template:Sfn

A subset of $C$ of a vector space $X$ is called a proper cone if it is a convex cone satisfying $C\cap (-C)=\{0\}.$ Explicitly, $C$ is a proper cone if (1) $C+C\subseteq C,$ (2) $rC\subseteq C$ for all $r>0,$ and (3) $C\cap (-C)=\{0\}.$Template:Sfn 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 $x\le y$ if and only if $y-x\in C,$ 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.Template:Sfn A total vector ordering cannot be Archimedean if its dimension, when considered as a vector space over the reals, is greater than 1.Template:Sfn

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 $P\subseteq Q.$Template:Sfn

Intervals and the order bound dual

An order interval in a preordered vector space is a set of the form $$\begin{array}{cc}[a,b]& =\{x:a\le x\le b\},\\ [a,b[& =\{x:a\le x<b\},\\ ]a,b]& =\{x:a<x\le b\},\text{ or }\\ ]a,b[& =\{x:a<x<b\}.\end{array}$$ From axioms 1 and 2 above it follows that $x,y\in [a,b]$ and $0<t<1$ implies $tx+(1-t)y$ belongs to $[a,b];$ thus these order intervals are convex. A subset is said to be order bounded if it is contained in some order interval.Template:Sfn In a preordered real vector space, if for $x\ge 0$ then the interval of the form $[-x,x]$ is balanced.Template:Sfn An order unit of a preordered vector space is any element $x$ such that the set $[-x,x]$ is absorbing.Template:Sfn

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 $X^{\mathrm{b}}.$Template:Sfn 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 $B\subseteq A$ such that $B$ is order bounded in $A,$ both $supB$ and $infB$ 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 $X.$Template:Sfn

Examples

If $(X,\le )$ is a preordered vector space over the reals with order unit $u,$ then the map $p(x):=inf\{t\in ℝ:x\le tu\}$ is a sublinear functional.Template:Sfn

Properties

If $X$ is a preordered vector space then for all $x,y\in X,$

• $x\ge 0$ and $y\ge 0$ imply $x+y\ge 0.$Template:Sfn
• $x\le y$ if and only if $-y\le -x.$Template:Sfn
• $x\le y$ and $r<0$ imply $rx\ge ry.$Template:Sfn
• $x\le y$ if and only if $y=sup\{x,y\}$ if and only if $x=inf\{x,y\}$Template:Sfn
• $sup\{x,y\}$ exists if and only if $inf\{-x,-y\}$ exists, in which case $inf\{-x,-y\}=-sup\{x,y\}.$Template:Sfn
• $sup\{x,y\}$ exists if and only if $inf\{x,y\}$ exists, in which case for all $z\in X,$Template:Sfn
  • $sup\{x+z,y+z\}=z+sup\{x,y\},$ and
  • $inf\{x+z,y+z\}=z+inf\{x,y\}$
  • $x+y=inf\{x,y\}+sup\{x,y\}.$
• $X$ is a vector lattice if and only if $sup\{0,x\}$ exists for all $x\in X.$Template:Sfn

Spaces of linear maps

A cone $C$ is said to be generating if $C-C$ is equal to the whole vector space.Template:Sfn 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 $C=\{u\in L(X;W):u(P)\subseteq Q\}$ is a proper cone in $L(X;W),$ 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 $L(X;W).$Template:Sfn More generally, if $M$ is any vector subspace of $L(X;W)$ such that $C\cap M$ is a proper cone, the ordering defined by $C\cap M$ is called the canonical ordering of $M.$Template:Sfn

Positive functionals and the order dual

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

1. $x\ge 0$ implies $f(x)\ge 0.$
2. if $x\le y$ then $f(x)\le f(y).$Template:Sfn

The set of all positive linear forms on a vector space with positive cone $C,$ called the dual cone and denoted by $C^{\ast },$ 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 Template:Visible anchor.Template:Sfn

The order dual of an ordered vector space $X$ is the set, denoted by $X^{+},$ defined by $X^{+}:=C^{\ast }-C^{\ast }.$ Although $X^{+}\subseteq X^b,$ there do exist ordered vector spaces for which set equality does Template:Em hold.Template:Sfn

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 $\{nx:n\in \mathbb{N}\}$ is majorized (that is, there exists some $y\in X$ such that $nx\le y$ for all $n\in \mathbb{N}$) then $x\le 0.$Template:Sfn A topological vector space (TVS) that is an ordered vector space is necessarily Archimedean if its positive cone is closed.Template:Sfn

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 $X.$Template:Sfn 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.Template:Sfn

An ordered vector space is called a vector lattice if for all elements $x$ and $y,$ the supremum $sup(x,y)$ and infimum $inf(x,y)$ exist.Template:Sfn

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 $C\cap M,$ where this cone is proper if $C$ is proper.Template:Sfn

Quotient space

Let $M$ be a vector subspace of an ordered vector space $X,$ $\pi :X\to X/M$ be the canonical projection, and let $\hat{C}:=\pi (C).$ Then $\hat{C}$ is a cone in $X/M$ that induces a canonical preordering on the quotient space $X/M.$ If $\hat{C}$ is a proper cone in$X/M$ then $\hat{C}$ makes $X/M$ into an ordered vector space.Template:Sfn If $M$ is $C$-saturated then $\hat{C}$ defines the canonical order of $X/M.$Template:Sfn Note that $X={ℝ}_0^2$ provides an example of an ordered vector space where $\pi (C)$ is not a proper cone.

If $X$ is also a topological vector space (TVS) and if for each neighborhood $V$ of the origin in $X$ there exists a neighborhood $U$ of the origin such that $[(U+N)\cap C]\subseteq V+N$ then $\hat{C}$ is a normal cone for the quotient topology.Template:Sfn

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.Template:Sfn

Product

If $S$ is any set then the space $X^S$ of all functions from $S$ into $X$ is canonically ordered by the proper cone $\{f\in X^S:f(s)\in C\text{ for all }s\in S\}.$Template:Sfn

Suppose that $\{X_{\alpha }:\alpha \in A\}$ is a family of preordered vector spaces and that the positive cone of $X_{\alpha }$ is $C_{\alpha }.$ Then $C:={\prod }_{\alpha }{C}_{\alpha }$ is a pointed convex cone in ${\prod }_{\alpha }{X}_{\alpha },$ which determines a canonical ordering on ${\prod }_{\alpha }{X}_{\alpha };$ $C$ is a proper cone if all $C_{\alpha }$ are proper cones.Template:Sfn

Algebraic direct sum

The algebraic direct sum ${⨁}_{\alpha }{X}_{\alpha }$ of $\{X_{\alpha }:\alpha \in A\}$ is a vector subspace of ${\prod }_{\alpha }{X}_{\alpha }$ that is given the canonical subspace ordering inherited from ${\prod }_{\alpha }{X}_{\alpha }.$Template:Sfn If $X_1,\dots ,X_n$ 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 $\prod _{\alpha }{X}_{\alpha }$ (with the canonical product order) is an order isomorphism.Template:Sfn

Examples

• The real numbers with the usual ordering form a totally ordered vector space. For all integers $n\ge 0,$ the Euclidean space ${ℝ}^n$ 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 $n=1$.Template:Sfn
• ${ℝ}^2$ is an ordered vector space with the $\le$ relation defined in any of the following ways (in order of increasing strength, that is, decreasing sets of pairs):
  • Lexicographical order: $(a,b)\le (c,d)$ if and only if $a<c$ or $(a=c\text{ and }b\le d).$ This is a total order. The positive cone is given by $x>0$ or $(x=0\text{ and }y\ge 0),$ that is, in polar coordinates, the set of points with the angular coordinate satisfying $-\pi /2<\theta \le \pi /2,$ together with the origin.
  • $(a,b)\le (c,d)$ if and only if $a\le c$ and $b\le d$ (the product order of two copies of $ℝ$ with $\le$). This is a partial order. The positive cone is given by $x\ge 0$ and $y\ge 0,$ that is, in polar coordinates $0\le \theta \le \pi /2,$ together with the origin.
  • $(a,b)\le (c,d)$ if and only if $(a<c\text{ and }b<d)$ or $(a=c\text{ and }b=d)$ (the reflexive closure of the direct product of two copies of $ℝ$ with "<"). This is also a partial order. The positive cone is given by $(x>0\text{ and }y>0)$ or $x=y=0),$ that is, in polar coordinates, $0<\theta <\pi /2,$ together with the origin.

Only the second order is, as a subset of ${ℝ}^4,$ closed; see partial orders in topological spaces.

For the third order the two-dimensional "intervals" $p<x<q$ are open sets which generate the topology.

• ${ℝ}^n$ is an ordered vector space with the $\le$ relation defined similarly. For example, for the second order mentioned above:
  • $x\le y$ if and only if $x_i\le y_i$ for $i=1,\dots ,n.$
• A Riesz space is an ordered vector space where the order gives rise to a lattice.
• The space of continuous functions on $[0,1]$ where $f\le g$ if and only if $f(x)\le g(x)$ for all $x$ in $[0,1].$
• Let ${\text{Sym}}_n(ℝ)$ denote the symmetric $n\times n$ matrices with real entries. The Loewner order $\preccurlyeq$ on two symmetric matrices $A,B\in {\text{Sym}}_n(ℝ)$is defined by $A\preccurlyeq B\iff B-A$ 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

If $S$ is any set and if $X$ is a vector space (over the reals) of real-valued functions on $S,$ then the pointwise order on $X$ is given by, for all $f,g\in X,$ $f\le g$ if and only if $f(s)\le g(s)$ for all $s\in S.$Template:Sfn

Spaces that are typically assigned this order include:

• the space ${\ell }^{\mathrm{\infty }}(S,ℝ)$ of bounded real-valued maps on $S.$
• the space $c_0(ℝ)$ of real-valued sequences that converge to $0.$
• the space $C(S,ℝ)$ of continuous real-valued functions on a topological space $S.$
• for any non-negative integer $n,$ the Euclidean space ${ℝ}^n$ when considered as the space $C(\{1,\dots ,n\},ℝ)$ where $S=\{1,\dots ,n\}$ is given the discrete topology.

The space ${ℒ}^{\mathrm{\infty }}(ℝ,ℝ)$ of all measurable almost-everywhere bounded real-valued maps on $ℝ,$ where the preorder is defined for all $f,g\in {ℒ}^{\mathrm{\infty }}(ℝ,ℝ)$ by $f\le g$ if and only if $f(s)\le g(s)$ almost everywhere.Template:Sfn

See also

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Bibliography

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• Bourbaki, Nicolas; Elements of Mathematics: Topological Vector Spaces; Template:Isbn.
• Template:Narici Beckenstein Topological Vector Spaces
• Template:Schaefer Wolff Topological Vector Spaces
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