Ordered vector space

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

1. $x\; \backslash leq\; y$ implies $x\; +\; z\; \backslash leq\; y\; +\; z,$
2. $y\; \backslash leq\; x$ implies $r\; y\; \backslash leq\; r\; x.$

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

Note that $x\; \backslash leq\; y$ if and only if $-y\; \backslash leq\; -x.$

A subset $C$ of a vector space $X$ is called a cone if for all real $r\; >\; 0,$ $r\; C\; \backslash subseteq\; C$. A cone is called pointed if it contains the origin. A cone $C$ is convex if and only if $C\; +\; C\; \backslash subseteq\; C.$ 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 $X\; =\; C\; -\; C.${{sfn|Schaefer|Wolff|1999|pp=250-257}}

Given a preordered vector space $X,$ the subset $X^+$ of all elements $x$ in $(X,\; \backslash leq)$ satisfying $x\; \backslash geq\; 0$ is a pointed convex cone (that is, a convex cone containing $0$) called the positive cone of $X$ and denoted by $\backslash operatorname\{PosCone\}\; X.$

The elements of the positive cone are called positive.

If $x$ and $y$ are elements of a preordered vector space $(X,\; \backslash leq),$ then $x\; \backslash leq\; y$ if and only if $y\; -\; x\; \backslash in\; X^+.$ The positive cone is generating if and only if $X$ is a directed set under $\backslash ,\backslash leq.$

Given any pointed convex cone $C$ one may define a preorder $\backslash ,\backslash leq\backslash ,$ on $X$ that is compatible with the vector space structure of $X$ by declaring for all $x,\; y\; \backslash in\; X,$ that $x\; \backslash leq\; y$ if and only if $y\; -\; x\; \backslash in\; C;$

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 $X.${{sfn|Schaefer|Wolff|1999|pp=250-257}}

If $X$ is preordered then we may form an equivalence relation on $X$ by defining $x$ is equivalent to $y$ if and only if $x\; \backslash leq\; y$ and $y\; \backslash leq\; x;$

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: $A\; \backslash leq\; B$ if and only there exist $a\; \backslash in\; A$ and $b\; \backslash in\; B$ such that $a\; \backslash leq\; b.${{sfn|Schaefer|Wolff|1999|pp=250-257}}

A subset of $C$ of a vector space $X$ is called a proper cone if it is a convex cone satisfying $C\; \backslash cap\; (-\; C)\; =\; \backslash \{0\backslash \}.$

Explicitly, $C$ is a proper cone if (1) $C\; +\; C\; \backslash subseteq\; C,$ (2) $r\; C\; \backslash subseteq\; C$ for all $r\; >\; 0,$ and (3) $C\; \backslash cap\; (-\; C)\; =\; \backslash \{0\backslash \}.${{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 $C$ in a real vector space induces an order on the vector space by defining $x\; \backslash leq\; y$ if and only if $y\; -\; x\; \backslash in\; C,$ 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.{{sfn|Schaefer|Wolff|1999|pp=250-257}}

A total vector ordering cannot be Archimedean if its dimension, when considered as a vector space over the reals, is greater than 1.{{sfn|Schaefer|Wolff|1999|pp=250-257}}

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 $P\; \backslash subseteq\; Q.${{sfn|Schaefer|Wolff|1999|pp=205–209}}

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

$$\backslash begin\{alignat\}\{4\}$$

[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}

From axioms 1 and 2 above it follows that $x,\; y\; \backslash in\; [a,\; b]$ and implies $t\; x\; +\; (1\; -\; t)\; y$ 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.{{sfn|Schaefer|Wolff|1999|pp=205–209}}

In a preordered real vector space, if for $x\; \backslash geq\; 0$ then the interval of the form $[-x,\; x]$ is balanced.{{sfn|Schaefer|Wolff|1999|pp=205–209}}

An order unit of a preordered vector space is any element $x$ such that the set $[-x,\; x]$ is absorbing.{{sfn|Schaefer|Wolff|1999|pp=205–209}}

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 $X^\{\backslash operatorname\{b\}\}.${{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 $A$ of an ordered vector space $X$ is called order complete if for every non-empty subset $B\; \backslash subseteq\; A$ such that $B$ is order bounded in $A,$ both $\backslash sup\; B$ and $\backslash inf\; B$ 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 $X.${{sfn|Schaefer|Wolff|1999|pp=204-214}}

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

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

• $x\; \backslash geq\; 0$ and $y\; \backslash geq\; 0$ imply $x\; +\; y\; \backslash geq\; 0.${{sfn|Narici|Beckenstein|2011|pp=139-153}}
• $x\; \backslash leq\; y$ if and only if $-y\; \backslash leq\; -x.${{sfn|Narici|Beckenstein|2011|pp=139-153}}
• $x\; \backslash leq\; y$ and imply $r\; x\; \backslash geq\; r\; y.${{sfn|Narici|Beckenstein|2011|pp=139-153}}
• $x\; \backslash leq\; y$ if and only if $y\; =\; \backslash sup\; \backslash \{x,\; y\backslash \}$ if and only if $x\; =\; \backslash inf\; \backslash \{x,\; y\backslash \}${{sfn|Narici|Beckenstein|2011|pp=139-153}}
• $\backslash sup\; \backslash \{x,\; y\backslash \}$ exists if and only if $\backslash inf\; \backslash \{-x,\; -y\backslash \}$ exists, in which case $\backslash inf\; \backslash \{-x,\; -y\backslash \}\; =\; -\; \backslash sup\; \backslash \{x,\; y\backslash \}.${{sfn|Narici|Beckenstein|2011|pp=139-153}}
• $\backslash sup\; \backslash \{x,\; y\backslash \}$ exists if and only if $\backslash inf\; \backslash \{x,\; y\backslash \}$ exists, in which case for all $z\; \backslash in\; X,${{sfn|Narici|Beckenstein|2011|pp=139-153}}
• $\backslash sup\; \backslash \{x\; +\; z,\; y\; +\; z\backslash \}\; =\; z\; +\; \backslash sup\; \backslash \{x,\; y\backslash \},$ and
• $\backslash inf\; \backslash \{x\; +\; z,\; y\; +\; z\backslash \}\; =\; z\; +\; \backslash inf\; \backslash \{x,\; y\backslash \}$
• $x\; +\; y\; =\; \backslash inf\backslash \{x,\; y\backslash \}\; +\; \backslash sup\; \backslash \{x,\; y\backslash \}.$
• $X$ is a vector lattice if and only if $\backslash sup\; \backslash \{0,\; x\backslash \}$ exists for all $x\; \backslash in\; X.${{sfn|Narici|Beckenstein|2011|pp=139-153}}

{{Main|Positive linear operator}}

A cone $C$ is said to be generating if $C\; -\; C$ is equal to the whole vector space.{{sfn|Schaefer|Wolff|1999|pp=205–209}}

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 $C\; =\; \backslash \{u\; \backslash in\; L(X;\; W)\; :\; u(P)\; \backslash subseteq\; Q\backslash \}$ 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 $L(X;\; W).${{sfn|Schaefer|Wolff|1999|pp=205–209}}

More generally, if $M$ is any vector subspace of $L(X;\; W)$ such that $C\; \backslash cap\; M$ is a proper cone, the ordering defined by $C\; \backslash cap\; M$ is called the canonical ordering of $M.${{sfn|Schaefer|Wolff|1999|pp=205–209}}

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

1. $x\; \backslash geq\; 0$ implies $f(x)\; \backslash geq\; 0.$
2. if $x\; \backslash leq\; y$ then $f(x)\; \backslash leq\; f(y).${{sfn|Narici|Beckenstein|2011|pp=139-153}}

The set of all positive linear forms on a vector space with positive cone $C,$ called the dual cone and denoted by $C^*,$ 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 {{visible anchor|dual preorder}}.{{sfn|Narici|Beckenstein|2011|pp=139-153}}

The order dual of an ordered vector space $X$ is the set, denoted by $X^+,$ defined by $X^+\; :=\; C^*\; -\; C^*.$

Although $X^+\; \backslash subseteq\; X^b,$ there do exist ordered vector spaces for which set equality does {{em|not}} hold.{{sfn|Schaefer|Wolff|1999|pp=205–209}}

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 $\backslash \{n\; x\; :\; n\; \backslash in\; \backslash N\backslash \}$ is majorized (that is, there exists some $y\; \backslash in\; X$ such that $n\; x\; \backslash leq\; y$ for all $n\; \backslash in\; \backslash N$) then $x\; \backslash leq\; 0.${{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 $X$ is regularly ordered and that its order is regular if it is Archimedean ordered and $X^+$ distinguishes points in $X.${{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 $x$ and $y,$ the supremum $\backslash sup\; (x,\; y)$ and infimum $\backslash inf\; (x,\; y)$ exist.{{sfn|Schaefer|Wolff|1999|pp=205–209}}

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 $C\; \backslash cap\; M,$ where this cone is proper if $C$ is proper.{{sfn|Schaefer|Wolff|1999|pp=205–209}}

Quotient space

Let $M$ be a vector subspace of an ordered vector space $X,$ $\backslash pi\; :\; X\; \backslash to\; X\; /\; M$ be the canonical projection, and let $\backslash hat\{C\}\; :=\; \backslash pi(C).$

Then $\backslash hat\{C\}$ is a cone in $X\; /\; M$ that induces a canonical preordering on the quotient space $X\; /\; M.$

If $\backslash hat\{C\}$ is a proper cone in$X\; /\; M$ then $\backslash hat\{C\}$ makes $X\; /\; M$ into an ordered vector space.{{sfn|Schaefer|Wolff|1999|pp=205–209}}

If $M$ is $C$-saturated then $\backslash hat\{C\}$ defines the canonical order of $X\; /\; M.${{sfn|Schaefer|Wolff|1999|pp=250-257}}

Note that $X\; =\; \backslash Reals^2\_0$ provides an example of an ordered vector space where $\backslash pi(C)$ 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 $[(U\; +\; N)\; \backslash cap\; C]\; \backslash subseteq\; V\; +\; N$ then $\backslash hat\{C\}$ is a normal cone for the quotient topology.{{sfn|Schaefer|Wolff|1999|pp=250-257}}

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.{{sfn|Schaefer|Wolff|1999|pp=250-257}}

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 $\backslash left\backslash \{f\; \backslash in\; X^S\; :\; f(s)\; \backslash in\; C\; \backslash text\{\; for\; all\; \}\; s\; \backslash in\; S\backslash right\backslash \}.${{sfn|Schaefer|Wolff|1999|pp=205–209}}

Suppose that $\backslash left\backslash \{X\_\backslash alpha\; :\; \backslash alpha\; \backslash in\; A\backslash right\backslash \}$ is a family of preordered vector spaces and that the positive cone of $X\_\backslash alpha$ is $C\_\backslash alpha.$

Then $C\; :=\; \backslash prod\_\backslash alpha\; C\_\backslash alpha$ is a pointed convex cone in $\backslash prod\_\backslash alpha\; X\_\backslash alpha,$ which determines a canonical ordering on $\backslash prod\_\backslash alpha\; X\_\backslash alpha;$

$C$ is a proper cone if all $C\_\backslash alpha$ are proper cones.{{sfn|Schaefer|Wolff|1999|pp=205–209}}

Algebraic direct sum

The algebraic direct sum $\backslash bigoplus\_\backslash alpha\; X\_\backslash alpha$ of $\backslash left\backslash \{X\_\backslash alpha\; :\; \backslash alpha\; \backslash in\; A\backslash right\backslash \}$ is a vector subspace of $\backslash prod\_\backslash alpha\; X\_\backslash alpha$ that is given the canonical subspace ordering inherited from $\backslash prod\_\backslash alpha\; X\_\backslash alpha.${{sfn|Schaefer|Wolff|1999|pp=205–209}}

If $X\_1,\; \backslash dots,\; X\_n$ 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 $\backslash prod\_\backslash alpha\; X\_\backslash alpha$ (with the canonical product order) is an order isomorphism.{{sfn|Schaefer|Wolff|1999|pp=205–209}}

• The real numbers with the usual ordering form a totally ordered vector space. For all integers $n\; \backslash geq\; 0,$ the Euclidean space $\backslash Reals^n$ 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 $n\; =\; 1$.{{sfn|Narici|Beckenstein|2011|pp=139-153}}
• $\backslash Reals^2$ is an ordered vector space with the $\backslash ,\backslash leq\backslash ,$ relation defined in any of the following ways (in order of increasing strength, that is, decreasing sets of pairs):
• Lexicographical order: $(a,\; b)\; \backslash leq\; (c,\; d)$ if and only if or $(a\; =\; c\; \backslash text\{\; and\; \}\; b\; \backslash leq\; d).$ This is a total order. The positive cone is given by $x\; >\; 0$ or $(x\; =\; 0\; \backslash text\{\; and\; \}\; y\; \backslash geq\; 0),$ that is, in polar coordinates, the set of points with the angular coordinate satisfying together with the origin.
• $(a,\; b)\; \backslash leq\; (c,\; d)$ if and only if $a\; \backslash leq\; c$ and $b\; \backslash leq\; d$ (the product order of two copies of $\backslash Reals$ with $\backslash leq$). This is a partial order. The positive cone is given by $x\; \backslash geq\; 0$ and $y\; \backslash geq\; 0,$ that is, in polar coordinates $0\; \backslash leq\; \backslash theta\; \backslash leq\; \backslash pi\; /\; 2,$ together with the origin.
• $(a,\; b)\; \backslash leq\; (c,\; d)$ if and only if or $(a\; =\; c\; \backslash text\{\; and\; \}\; b\; =\; d)$ (the reflexive closure of the direct product of two copies of $\backslash Reals$ with "<"). This is also a partial order. The positive cone is given by $(x\; >\; 0\; \backslash text\{\; and\; \}\; y\; >\; 0)$ or $x\; =\; y\; =\; 0),$ that is, in polar coordinates, together with the origin.

:Only the second order is, as a subset of $\backslash Reals^4,$ closed; see partial orders in topological spaces.

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

• $\backslash Reals^n$ is an ordered vector space with the $\backslash ,\backslash leq\backslash ,$ relation defined similarly. For example, for the second order mentioned above:
• $x\; \backslash leq\; y$ if and only if $x\_i\; \backslash leq\; y\_i$ for $i\; =\; 1,\; \backslash dots,\; n.$
• 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 $f\; \backslash leq\; g$ if and only if $f(x)\; \backslash leq\; g(x)$ for all $x$ in $[0,\; 1].$

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 $f,\; g\; \backslash in\; X,$ $f\; \backslash leq\; g$ if and only if $f(s)\; \backslash leq\; g(s)$ for all $s\; \backslash in\; S.${{sfn|Narici|Beckenstein|2011|pp=139-153}}

Spaces that are typically assigned this order include:

• the space $\backslash ell^\backslash infty(S,\; \backslash Reals)$ of bounded real-valued maps on $S.$
• the space $c\_0(\backslash Reals)$ of real-valued sequences that converge to $0.$
• the space $C(S,\; \backslash Reals)$ of continuous real-valued functions on a topological space $S.$
• for any non-negative integer $n,$ the Euclidean space $\backslash Reals^n$ when considered as the space $C(\backslash \{1,\; \backslash dots,\; n\backslash \},\; \backslash Reals)$ where $S\; =\; \backslash \{1,\; \backslash dots,\; n\backslash \}$ is given the discrete topology.

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

• {{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}}

{{reflist}}

• {{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; Elements of Mathematics: Topological Vector Spaces; {{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