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
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