8

1. Cones

Cartesian product L = Yliei Li, called the product cone of the family

of cones {L+}iG/, and so (L,L

+

) is an ordered vector space. Notice that

(%i)iei {l/i)i£i holds in L if and only if X{ yi holds in Li for each i. An

important particular case of a Cartesian product ordered vector space is the

ordered vector space MP, where Q is a nonempty set—and, as above, f g

holds in

Rn

if and only if f(u) g(u) for each u G Q. More generally, if L is

an ordered vector space and I is any nonempty index set, then the Cartesian

product

L1

is likewise an ordered vector space, where (xi)iei (yi)iei in

L1

is equivalent to Xi yi in L for each i e I.

Finally, we shall close the section with the notion of an order isomor-

phism. Recall that a function T: X — Y is called an operator (or a linear

transformation) if T{ax + j3y) = aT(x) + (3T(y) holds for all x, y e X

and all scalars a, (3 G R.

Definition 1.9. Let L and M be two ordered vector spaces. An operator

T: L — M is called an order isomorphism (or an order-embedding) if

(a) T is one-to-one and

(b) x 0 holds in L if and only ifTx 0 holds in M.

If there exists an onto (i.e., surjective) order isomorphism from L to M,

then L and M are called order isomorphic ordered vector spaces?

That is, two ordered vector spaces are order isomorphic if and only if

there exists a one-to-one correspondence between the elements of the ordered

vector spaces that preserves both the algebraic and the order structures.

From the point of view of the theory of ordered vector spaces two order

isomorphic ordered vector spaces are viewed as identical objects.

We shall also say that an ordered vector space L is order-embeddable

into another ordered vector space M if there exists an order isomorphism

from L to M. In this case, the subspace T(L) ordered with the induced

vector ordering from M can be identified with L and for this reason T(L) is

referred to as a copy of L in M.

Exercises

1. Show that a nonempty convex subset of a vector space is a wedge if and

only if it is closed under nonnegative scalar multiplication.

2. If W is a wedge, then prove that W fl (—W) is a vector subspace.

3. If two vectors x and y of a cone satisfy x-\-y = 0, then show that x = y = 0.

Since the structures under consideration here are the vector orderings, according to the

general notion of an isomorphism (as explained following the Preface of the book) the "order

isomorphism" can be very well referred to simply as an "isomorphism."