Ordered vector space

A point ${\displaystyle x}$ in ${\displaystyle \mathbb {R} ^{2}}$ and the set of all ${\displaystyle y}$ such that ${\displaystyle x\leq y}$ (in red). The order here is ${\displaystyle x\leq y}$ if and only if ${\displaystyle x_{1}\leq y_{1}}$ and ${\displaystyle x_{2}\leq y_{2}.}$

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

1. ${\displaystyle x\leq y}$ implies ${\displaystyle x+z\leq y+z,}$
2. ${\displaystyle y\leq x}$ implies ${\displaystyle ry\leq rx.}$

If ${\displaystyle \,\leq \,}$ is a partial order compatible with the vector space structure of ${\displaystyle X}$ then ${\displaystyle (X,\leq )}$ is called an ordered vector space and ${\displaystyle \,\leq \,}$ is called a vector partial order on ${\displaystyle X.}$ The two axioms imply that translations and positive homotheties are automorphisms of the order structure and the mapping ${\displaystyle x\mapsto -x}$ is an isomorphism to the dual order structure. Ordered vector spaces are ordered groups under their addition operation. Note that ${\displaystyle x\leq y}$ if and only if ${\displaystyle -y\leq -x.}$

Positive cones and their equivalence to orderings

A subset ${\displaystyle C}$ of a vector space ${\displaystyle X}$ is called a cone if for all real ${\displaystyle r>0,}$${\displaystyle rC\subseteq C.}$ A cone is called pointed if it contains the origin. A cone ${\displaystyle C}$ is convex if and only if ${\displaystyle C+C\subseteq C.}$ 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 ${\displaystyle C}$ in a vector space ${\displaystyle X}$ is said to be generating if ${\displaystyle X=C-C.}$ ^[1]

Given a preordered vector space ${\displaystyle X,}$ the subset ${\displaystyle X^{+}}$ of all elements ${\displaystyle x}$ in ${\displaystyle (X,\leq )}$ satisfying ${\displaystyle x\geq 0}$ is a pointed convex cone with vertex ${\displaystyle 0}$ (that is, it contains ${\displaystyle 0}$) called the positive cone of ${\displaystyle X}$ and denoted by ${\displaystyle \operatorname {PosCone} X.}$ The elements of the positive cone are called positive. If ${\displaystyle x}$ and ${\displaystyle y}$ are elements of a preordered vector space ${\displaystyle (X,\leq ),}$ then ${\displaystyle x\leq y}$ if and only if ${\displaystyle y-x\in X^{+}.}$ The positive cone is generating if and only if ${\displaystyle X}$ is a directed set under ${\displaystyle \,\leq .}$ Given any pointed convex cone ${\displaystyle C}$ with vertex ${\displaystyle 0,}$ one may define a preorder ${\displaystyle \,\leq \,}$ on ${\displaystyle X}$ that is compatible with the vector space structure of ${\displaystyle X}$ by declaring for all ${\displaystyle x,y\in X,}$ that ${\displaystyle x\leq y}$ if and only if ${\displaystyle y-x\in C;}$ the positive cone of this resulting preordered vector space is ${\displaystyle C.}$ There is thus a one-to-one correspondence between pointed convex cones with vertex ${\displaystyle 0}$ and vector preorders on ${\displaystyle X.}$ ^[1] If ${\displaystyle X}$ is preordered then we may form an equivalence relation on ${\displaystyle X}$ by defining ${\displaystyle x}$ is equivalent to ${\displaystyle y}$ if and only if ${\displaystyle x\leq y}$ and ${\displaystyle y\leq x;}$ if ${\displaystyle N}$ is the equivalence class containing the origin then ${\displaystyle N}$ is a vector subspace of ${\displaystyle X}$ and ${\displaystyle X/N}$ is an ordered vector space under the relation: ${\displaystyle A\leq B}$ if and only there exist ${\displaystyle a\in A}$ and ${\displaystyle b\in B}$ such that ${\displaystyle a\leq b.}$ ^[1]

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

By a total vector ordering on ${\displaystyle X}$ we mean a total order on ${\displaystyle X}$ that is compatible with the vector space structure of ${\displaystyle X.}$ The family of total vector orderings on a vector space ${\displaystyle X}$ 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 ${\displaystyle R}$ and ${\displaystyle S}$ are two orderings of a vector space with positive cones ${\displaystyle P}$ and ${\displaystyle Q,}$ respectively, then we say that ${\displaystyle R}$ is finer than ${\displaystyle S}$ if ${\displaystyle P\subseteq Q.}$ ^[2]

Examples

The real numbers with the usual ordering form a totally ordered vector space. For all integers ${\displaystyle n\geq 0,}$ the Euclidean space ${\displaystyle \mathbb {R} ^{n}}$ 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 ${\displaystyle n=1}$. ^[3]

Pointwise order

If ${\displaystyle S}$ is any set and if ${\displaystyle X}$ is a vector space (over the reals) of real-valued functions on ${\displaystyle S,}$ then the pointwise order on ${\displaystyle X}$ is given by, for all ${\displaystyle f,g\in X,}$${\displaystyle f\leq g}$ if and only if ${\displaystyle f(s)\leq g(s)}$ for all ${\displaystyle s\in S.}$ ^[3]

Spaces that are typically assigned this order include:

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

The space ${\displaystyle {\mathcal {L}}^{\infty }(\mathbb {R} ,\mathbb {R} )}$ of all measurable almost-everywhere bounded real-valued maps on ${\displaystyle \mathbb {R} ,}$ where the preorder is defined for all ${\displaystyle f,g\in {\mathcal {L}}^{\infty }(\mathbb {R} ,\mathbb {R} )}$ by ${\displaystyle f\leq g}$ if and only if ${\displaystyle f(s)\leq g(s)}$ almost everywhere. ^[3]

Intervals and the order bound dual

An order interval in a preordered vector space is set of the form

$${\displaystyle {\begin{alignedat}{4}[a,b]&=\{x:a\leq x\leq b\},\\[0.1ex][a,b[&=\{x:a\leq x<b\},\\]a,b]&=\{x:a<x\leq b\},{\text{ or }}\\]a,b[&=\{x:a<x<b\}.\\\end{alignedat}}}$$

From axioms 1 and 2 above it follows that ${\displaystyle x,y\in [a,b]}$ and ${\displaystyle 0<t<1}$ implies ${\displaystyle tx+(1-t)y}$ belongs to ${\displaystyle [a,b];}$ 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 ${\displaystyle x\geq 0}$ then the interval of the form ${\displaystyle [-x,x]}$ is balanced. ^[2] An order unit of a preordered vector space is any element ${\displaystyle x}$ such that the set ${\displaystyle [-x,x]}$ is absorbing. ^[2]

The set of all linear functionals on a preordered vector space ${\displaystyle X}$ that map every order interval into a bounded set is called the order bound dual of ${\displaystyle X}$ and denoted by ${\displaystyle X^{\operatorname {b} }.}$ ^[2] If a space is ordered then its order bound dual is a vector subspace of its algebraic dual.

A subset ${\displaystyle A}$ of an ordered vector space ${\displaystyle X}$ is called order complete if for every non-empty subset ${\displaystyle B\subseteq A}$ such that ${\displaystyle B}$ is order bounded in ${\displaystyle A,}$ both ${\displaystyle \sup B}$ and ${\displaystyle \inf B}$ exist and are elements of ${\displaystyle A.}$ We say that an ordered vector space ${\displaystyle X}$ is order complete is ${\displaystyle X}$ is an order complete subset of ${\displaystyle X.}$ ^[4]

Examples

If ${\displaystyle (X,\leq )}$ is a preordered vector space over the reals with order unit ${\displaystyle u,}$ then the map ${\displaystyle p(x):=\inf\{t\in \mathbb {R} :x\leq tu\}}$ is a sublinear functional. ^[3]

Properties

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

• ${\displaystyle x\geq 0}$ and ${\displaystyle y\geq 0}$ imply ${\displaystyle x+y\geq 0.}$ ^[3]
• ${\displaystyle x\leq y}$ if and only if ${\displaystyle -y\leq -x.}$ ^[3]
• ${\displaystyle x\leq y}$ and ${\displaystyle r<0}$ imply ${\displaystyle rx\geq ry.}$ ^[3]
• ${\displaystyle x\leq y}$ if and only if ${\displaystyle y=\sup\{x,y\}}$ if and only if ${\displaystyle x=\inf\{x,y\}}$ ^[3]
• ${\displaystyle \sup\{x,y\}}$ exists if and only if ${\displaystyle \inf\{-x,-y\}}$ exists, in which case ${\displaystyle \inf\{-x,-y\}=-\sup\{x,y\}.}$ ^[3]
• ${\displaystyle \sup\{x,y\}}$ exists if and only if ${\displaystyle \inf\{x,y\}}$ exists, in which case for all ${\displaystyle z\in X,}$ ^[3]
  • ${\displaystyle \sup\{x+z,y+z\}=z+\sup\{x,y\},}$ and
  • ${\displaystyle \inf\{x+z,y+z\}=z+\inf\{x,y\}}$
  • ${\displaystyle x+y=\inf\{x,y\}+\sup\{x,y\}.}$
• ${\displaystyle X}$ is a vector lattice if and only if ${\displaystyle \sup\{0,x\}}$ exists for all ${\displaystyle x\in X.}$ ^[3]

Spaces of linear maps

Main article: Positive linear operator

A cone ${\displaystyle C}$ is said to be generating if ${\displaystyle C-C}$ is equal to the whole vector space. ^[2] If ${\displaystyle X}$ and ${\displaystyle W}$ are two non-trivial ordered vector spaces with respective positive cones ${\displaystyle P}$ and ${\displaystyle Q,}$ then ${\displaystyle P}$ is generating in ${\displaystyle X}$ if and only if the set ${\displaystyle C=\{u\in L(X;W):u(P)\subseteq Q\}}$ is a proper cone in ${\displaystyle L(X;W),}$ which is the space of all linear maps from ${\displaystyle X}$ into ${\displaystyle W.}$ In this case, the ordering defined by ${\displaystyle C}$ is called the canonical ordering of ${\displaystyle L(X;W).}$ ^[2] More generally, if ${\displaystyle M}$ is any vector subspace of ${\displaystyle L(X;W)}$ such that ${\displaystyle C\cap M}$ is a proper cone, the ordering defined by ${\displaystyle C\cap M}$ is called the canonical ordering of ${\displaystyle M.}$ ^[2]

Positive functionals and the order dual

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

1. ${\displaystyle x\geq 0}$ implies ${\displaystyle f(x)\geq 0.}$
2. if ${\displaystyle x\leq y}$ then ${\displaystyle f(x)\leq f(y).}$ ^[3]

The set of all positive linear forms on a vector space with positive cone ${\displaystyle C,}$ called the dual cone and denoted by ${\displaystyle C^{*},}$ is a cone equal to the polar of ${\displaystyle -C.}$ The preorder induced by the dual cone on the space of linear functionals on ${\displaystyle X}$ is called the dual preorder. ^[3]

The order dual of an ordered vector space ${\displaystyle X}$ is the set, denoted by ${\displaystyle X^{+},}$ defined by ${\displaystyle X^{+}:=C^{*}-C^{*}.}$ Although ${\displaystyle X^{+}\subseteq X^{b},}$ there do exist ordered vector spaces for which set equality does not hold. ^[2]

Special types of ordered vector spaces

Let ${\displaystyle X}$ be an ordered vector space. We say that an ordered vector space ${\displaystyle X}$ is Archimedean ordered and that the order of ${\displaystyle X}$ is Archimedean if whenever ${\displaystyle x}$ in ${\displaystyle X}$ is such that ${\displaystyle \{nx:n\in \mathbb {N} \}}$ is majorized (that is, there exists some ${\displaystyle y\in X}$ such that ${\displaystyle nx\leq y}$ for all ${\displaystyle n\in \mathbb {N} }$) then ${\displaystyle x\leq 0.}$ ^[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 ${\displaystyle X}$ is regularly ordered and that its order is regular if it is Archimedean ordered and ${\displaystyle X^{+}}$ distinguishes points in ${\displaystyle X.}$ ^[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 ${\displaystyle x}$ and ${\displaystyle y,}$ the supremum ${\displaystyle \sup(x,y)}$ and infimum ${\displaystyle \inf(x,y)}$ exist. ^[2]

Subspaces, quotients, and products

Throughout let ${\displaystyle X}$ be a preordered vector space with positive cone ${\displaystyle C.}$

Subspaces

If ${\displaystyle M}$ is a vector subspace of ${\displaystyle X}$ then the canonical ordering on ${\displaystyle M}$ induced by ${\displaystyle X}$'s positive cone ${\displaystyle C}$ is the partial order induced by the pointed convex cone ${\displaystyle C\cap M,}$ where this cone is proper if ${\displaystyle C}$ is proper. ^[2]

Quotient space

Let ${\displaystyle M}$ be a vector subspace of an ordered vector space ${\displaystyle X,}$${\displaystyle \pi :X\to X/M}$ be the canonical projection, and let ${\displaystyle {\hat {C}}:=\pi (C).}$ Then ${\displaystyle {\hat {C}}}$ is a cone in ${\displaystyle X/M}$ that induces a canonical preordering on the quotient space ${\displaystyle X/M.}$ If ${\displaystyle {\hat {C}}}$ is a proper cone in${\displaystyle X/M}$ then ${\displaystyle {\hat {C}}}$ makes ${\displaystyle X/M}$ into an ordered vector space. ^[2] If ${\displaystyle M}$ is ${\displaystyle C}$-saturated then ${\displaystyle {\hat {C}}}$ defines the canonical order of ${\displaystyle X/M.}$ ^[1] Note that ${\displaystyle X=\mathbb {R} _{0}^{2}}$ provides an example of an ordered vector space where ${\displaystyle \pi (C)}$ is not a proper cone.

If ${\displaystyle X}$ is also a topological vector space (TVS) and if for each neighborhood ${\displaystyle V}$ of the origin in ${\displaystyle X}$ there exists a neighborhood ${\displaystyle U}$ of the origin such that ${\displaystyle [(U+N)\cap C]\subseteq V+N}$ then ${\displaystyle {\hat {C}}}$ is a normal cone for the quotient topology. ^[1]

If ${\displaystyle X}$ is a topological vector lattice and ${\displaystyle M}$ is a closed solid sublattice of ${\displaystyle X}$ then ${\displaystyle X/L}$ is also a topological vector lattice. ^[1]

Product

If ${\displaystyle S}$ is any set then the space ${\displaystyle X^{S}}$ of all functions from ${\displaystyle S}$ into ${\displaystyle X}$ is canonically ordered by the proper cone ${\displaystyle \left\{f\in X^{S}:f(s)\in C{\text{ for all }}s\in S\right\}.}$ ^[2]

Suppose that ${\displaystyle \left\{X_{\alpha }:\alpha \in A\right\}}$ is a family of preordered vector spaces and that the positive cone of ${\displaystyle X_{\alpha }}$ is ${\displaystyle C_{\alpha }.}$ Then ${\textstyle C:=\prod _{\alpha }C_{\alpha }}$ is a pointed convex cone in ${\textstyle \prod _{\alpha }X_{\alpha },}$ which determines a canonical ordering on ${\textstyle \prod _{\alpha }X_{\alpha };}$${\displaystyle C}$ is a proper cone if all ${\displaystyle C_{\alpha }}$ are proper cones. ^[2]

Algebraic direct sum

The algebraic direct sum ${\textstyle \bigoplus _{\alpha }X_{\alpha }}$ of ${\displaystyle \left\{X_{\alpha }:\alpha \in A\right\}}$ is a vector subspace of ${\textstyle \prod _{\alpha }X_{\alpha }}$ that is given the canonical subspace ordering inherited from ${\textstyle \prod _{\alpha }X_{\alpha }.}$ ^[2] If ${\displaystyle X_{1},\dots ,X_{n}}$ are ordered vector subspaces of an ordered vector space ${\displaystyle X}$ then ${\displaystyle X}$ is the ordered direct sum of these subspaces if the canonical algebraic isomorphism of ${\displaystyle X}$ onto ${\displaystyle \prod _{\alpha }X_{\alpha }}$ (with the canonical product order) is an order isomorphism. ^[2]

Examples

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

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

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

• ${\displaystyle \mathbb {R} ^{n}}$ is an ordered vector space with the ${\displaystyle \,\leq \,}$ relation defined similarly. For example, for the second order mentioned above:
  • ${\displaystyle x\leq y}$ if and only if ${\displaystyle x_{i}\leq y_{i}}$ for ${\displaystyle i=1,\dots ,n.}$
• A Riesz space is an ordered vector space where the order gives rise to a lattice.
• The space of continuous functions on ${\displaystyle [0,1]}$ where ${\displaystyle f\leq g}$ if and only if ${\displaystyle f(x)\leq g(x)}$ for all ${\displaystyle x}$ in ${\displaystyle [0,1].}$

See also

• 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 orderPages displaying short descriptions of redirect targets
• Ordered ring  – ring with a compatible total orderPages displaying wikidata descriptions as a fallback
• 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 latticePages displaying short descriptions of redirect targets

References

1. Schaefer & Wolff 1999, pp. 250–257.
2. Schaefer & Wolff 1999, pp. 205–209.
3. Narici & Beckenstein 2011, pp. 139–153.
4. Schaefer & Wolff 1999, pp. 204–214.

Bibliography

• 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.