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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