Ordered vector space explained

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

X

over the real numbers

\Reals

and a preorder

\leq

on the set

X,

the pair

(X,\leq)

is called a preordered vector space and we say that the preorder

\leq

is compatible with the vector space structure of

X

and call

\leq

a vector preorder on

X

if for all

x,y,z\inX

and

r\in\Reals

with

r\geq0

the following two axioms are satisfied

x\leqy

implies

x+z\leqy+z,

y\leqx

implies

ry\leqrx.

If

\leq

is a partial order compatible with the vector space structure of

X

then

(X,\leq)

is called an ordered vector space and

\leq

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\leqy

if and only if

-y\leq-x.

Positive cones and their equivalence to orderings

C

of a vector space

X

is called a cone if for all real

r>0,

rC\subseteqC,

that is, for all

c,c'\inC

we have

c+c'\inC

. A cone is called pointed if it contains the origin. A cone

C

is convex if and only if

C+C\subseteqC.

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.

Given a preordered vector space

X,

the subset

X+

of all elements

x

in

(X,\leq)

satisfying

x\geq0

is a pointed convex cone (that is, a convex cone containing

0

) called the positive cone of

X

and denoted by

\operatorname{PosCone}X.

The elements of the positive cone are called positive. If

x

and

y

are elements of a preordered vector space

(X,\leq),

then

x\leqy

if and only if

y-x\inX+.

The positive cone is generating if and only if

X

is a directed set under

\leq.

Given any pointed convex cone

C

one may define a preorder

\leq

on

X

that is compatible with the vector space structure of

X

by declaring for all

x,y\inX,

that

x\leqy

if and only if

y-x\inC;

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.

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\leqy

and

y\leqx;

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\leqB

if and only there exist

a\inA

and

b\inB

such that

a\leqb.

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\subseteqC,

(2)

rC\subseteqC

for all

r>0,

and (3)

C\cap(-C)=\{0\}.

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\leqy

if and only if

y-x\inC,

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. A total vector ordering cannot be Archimedean if its dimension, when considered as a vector space over the reals, is greater than 1.

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\subseteqQ.

Examples

The real numbers with the usual ordering form a totally ordered vector space. For all integers

n\geq0,

the Euclidean space

\Realsn

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

.

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\inX,

f\leqg

if and only if

f(s)\leqg(s)

for all

s\inS.

Spaces that are typically assigned this order include:

\ellinfty(S,\Reals)

of bounded real-valued maps on

S.

c0(\Reals)

of real-valued sequences that converge to

0.

C(S,\Reals)

of continuous real-valued functions on a topological space

S.

n,

the Euclidean space

\Realsn

when considered as the space

C(\{1,...,n\},\Reals)

where

S=\{1,...,n\}

is given the discrete topology.

The space

l{L}infty(\Reals,\Reals)

of all measurable almost-everywhere bounded real-valued maps on

\Reals,

where the preorder is defined for all

f,g\inl{L}infty(\Reals,\Reals)

by

f\leqg

if and only if

f(s)\leqg(s)

almost everywhere.

Intervals and the order bound dual

An order interval in a preordered vector space is set of the form \begin[a, b] &= \, \\[0.1ex][a, b[ &= \{x : a \leq x < b\}, \\ ]a, b] &= \, \text \\]a, b[&= \{x : a < x < b\}. \\ \end{alignat}</math> From axioms 1 and 2 above it follows that <math>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. In a preordered real vector space, if for

x\geq0

then the interval of the form

[-x,x]

is balanced. An order unit of a preordered vector space is any element

x

such that the set

[-x,x]

is absorbing.

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\operatorname{b

}. 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\subseteqA

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.

Examples

If

(X,\leq)

is a preordered vector space over the reals with order unit

u,

then the map

p(x):=inf\{t\in\Reals:x\leqtu\}

is a sublinear functional.

Properties

If

X

is a preordered vector space then for all

x,y\inX,

x\geq0

and

y\geq0

imply

x+y\geq0.

x\leqy

if and only if

-y\leq-x.

x\leqy

and

r<0

imply

rx\geqry.

x\leqy

if and only if

y=\sup\{x,y\}

if and only if

x=inf\{x,y\}

\sup\{x,y\}

exists if and only if

inf\{-x,-y\}

exists, in which case

inf\{-x,-y\}=-\sup\{x,y\}.

\sup\{x,y\}

exists if and only if

inf\{x,y\}

exists, in which case for all

z\inX,

\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\inX.

Spaces of linear maps

See main article: Positive linear operator.

A cone

C

is said to be generating if

C-C

is equal to the whole vector space. 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\inL(X;W):u(P)\subseteqQ\}

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).

More generally, if

M

is any vector subspace of

L(X;W)

such that

C\capM

is a proper cone, the ordering defined by

C\capM

is called the canonical ordering of

M.

Positive functionals and the order dual

f

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

x\geq0

implies

f(x)\geq0.

  1. if

x\leqy

then

f(x)\leqf(y).

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 .

The order dual of an ordered vector space

X

is the set, denoted by

X+,

defined by

X+:=C*-C*.

Although

X+\subseteqXb,

there do exist ordered vector spaces for which set equality does hold.

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\N\}

is majorized (that is, there exists some

y\inX

such that

nx\leqy

for all

n\in\N

) then

x\leq0.

A topological vector space (TVS) that is an ordered vector space is necessarily Archimedean if its positive cone is closed.

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.

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.

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.

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\capM,

where this cone is proper if

C

is proper.

Quotient space

Let

M

be a vector subspace of an ordered vector space

X,

\pi:X\toX/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. If

M

is

C

-saturated
then

\hat{C}

defines the canonical order of

X/M.

Note that

X=

2
\Reals
0
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)\capC]\subseteqV+N

then

\hat{C}

is a normal cone for the quotient topology.

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.

Product

If

S

is any set then the space

XS

of all functions from

S

into

X

is canonically ordered by the proper cone

\left\{f\inXS:f(s)\inCforalls\inS\right\}.

Suppose that

\left\{X\alpha:\alpha\inA\right\}

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.

Algebraic direct sum

The algebraic direct sum \bigoplus_\alpha X_\alpha of

\left\{X\alpha:\alpha\inA\right\}

is a vector subspace of \prod_\alpha X_\alpha that is given the canonical subspace ordering inherited from \prod_\alpha X_\alpha.If

X1,...,Xn

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\alphaX\alpha

(with the canonical product order) is an order isomorphism.

Examples

\Reals2

is an ordered vector space with the

\leq

relation defined in any of the following ways (in order of increasing strength, that is, decreasing sets of pairs):

(a,b)\leq(c,d)

if and only if

a<c

or

(a=candb\leqd).

This is a total order. The positive cone is given by

x>0

or

(x=0andy\geq0),

that is, in polar coordinates, the set of points with the angular coordinate satisfying

-\pi/2<\theta\leq\pi/2,

together with the origin.

(a,b)\leq(c,d)

if and only if

a\leqc

and

b\leqd

(the product order of two copies of

\Reals

with

\leq

). This is a partial order. The positive cone is given by

x\geq0

and

y\geq0,

that is, in polar coordinates

0\leq\theta\leq\pi/2,

together with the origin.

(a,b)\leq(c,d)

if and only if

(a<candb<d)

or

(a=candb=d)

(the reflexive closure of the direct product of two copies of

\Reals

with "<"). This is also a partial order. The positive cone is given by

(x>0andy>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

\Reals4,

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.

\Realsn

is an ordered vector space with the

\leq

relation defined similarly. For example, for the second order mentioned above:

x\leqy

if and only if

xi\leqyi

for

i=1,...,n.

[0,1]

where

f\leqg

if and only if

f(x)\leqg(x)

for all

x

in

[0,1].

Bibliography

. Charalambos D. Aliprantis. Burkinshaw, Owen. Locally solid Riesz spaces with applications to economics. Second. Providence, R. I.: American Mathematical Society. 2003. 0-8218-3408-8.