1.1. Wedges and cones 9

4. If Wi and W2 are two wedges of a vector space, then show that AWi for

A G R, Wi H VV2, and Wi + W2 are all wedges. Is the sum of two cones a

cone?

5. If two cones /Ci and /C2 of a vector space satisfy K\ f) (—JC2) — {0}, then

show that JC± + /C2 is a cone.

6. Prove that if K is a cone and W is a wedge of a vector space, then /C fl W

is a cone—and so, in particular, the intersection of two cones is a cone.

7. Consider the wedge W of R2 defined by W = {(x, y) G R2: x + y 0}.

According to Lemma 1.4 we know that W has a decomposition of the

form W = V + /C, where V is a vector subspace and /C is a cone. Find

three different such decompositions of W.

8. Establish that if e is an order unit of an ordered vector space L, then so

are e + x and Ae for all x G L+ and A 0.

9. Following the terminology in this section, we shall say that a vector sub-

space X of an ordered vector space L is majorizing if for each u G L

there exists some x G X such that u x. (A majorizing vector subspace

is also called a cofinal vector subspace.)

For an ordered vector space L establish the following.

(a) A vector subspace X of L majorizes L if and only if L = X — L+.

(b) The cone L

+

is generating if and only if the vector space L

+

— L+

majorizes L.

(c) If a vector subspace contains an order unit, then it majorizes L.

(d) A vector subspace that majorizes L need not have an order unit.

10. For an operator T: X — Y between vector spaces show the following.

(a) If W is a wedge of X, then T(W) is a wedge of Y.

(b) If W is a wedge of Y, then

T~l(W)

is a wedge of X.

(c) If T is one-to-one and W is a cone of X, then T(W) is a cone of Y.

(d) If T is one-to-one and W is a cone of Y, then

T~l(W)

is a cone of X.

11. Prove Lemma 1.4.

12. Can a cone contain a straight line?

13. Establish that a positive vector e of an ordered vector space is an order

unit if and only if the vector zero is an internal point of the convex set

e — L

+

= {x G L: x e}. [Hint: Notice that e is an internal point of L

+

if and only if 0 is an internal point of e — L

+

and then apply Lemma 1.7. ]

14. If [a, b] and [c, d] are order intervals of an ordered vector space, then

establish that [a, b] -f [c, d] = [a + c, b + d], A[a, 6] = [Aa, A6] for all A 0,

and A [a, b] = [A6, Aa] for all A 0.

15. Prove that translations, arbitrary intersections, and scalar multiples of

full sets are full sets.

16. Show that if B is a nonempty subset of an ordered vector space, then its

full hull [B] is given by

[B] = (B +

L+)n(B-L+)=

(J [x,y].

x,yEB