10 1. Cones

Use the formula [B] = (B + L+) n ( 5 - L+) to conclude that

(i) the full hull of a nonempty convex set is convex and

(ii) the full hull of a circled set is circled.

17. Let B be a nonempty subset of an ordered vector space consisting of

"incomparable" vectors, i.e, if x, y G B and x / y, then neither x y nor

y x is true. Find the full hull of B. [Hint: Note that [x,x] — {x}.]

18. If a nonempty subset A of an ordered vector space L has a supremum

(resp. an innmum), then show that for each x G L the set x -f A has a

supremum (resp. an infimum) and the following lattice identities hold:

x -f sup A = sup(x + A) and x + 'miA = inf (x + A).

Similarly, with an appropriate interpretation establish the following useful

lattice identities:

x — sup A — inf (x — A) and x — inf A — sup(x — A).

A sup A = sup(AA) and A inf A = inf (AA) for all A 0.

19. Assume that two non-empty subsets A and £? of an ordered vector space

have suprema. Then show that the set A + B also has a supremum and

that sup (A + B) = sup A -f sup £? holds true.

20. For each n G N let fn: [0,1] — R be the continuous function whose graph

consists of the two line segments joining the points (0,0) with (-, 1) and

(^,1) with (1,1). Show that fn ] 1 holds in C[0,1], where 1 denotes the

constant function one on [0,1].

Likewise, if {gn} Q C[0,1] is defined by gn(t) — y/i, then show that

0

n

T l i n C [ O , l ] .

21. For each n G N let hn : [0,1] — R be the continuous function whose graph

consists of the polygonal line joining the points (0,0), (^, 0), (\ + ^ , 1),

and (1,1). Show that hn | holds true in C[0,1] and that the sequence

{hn} does not have a supremum in C[0,1].

22. Show that every bounded from above nonempty subset A of RQ, where

£1 is a fixed nonempty set, has a supremum in the ordered vector space

R n and that [supA](cj) = sup^GA f(uj) holds for all u G ft.7

Use this conclusion to find the suprema of the sequences appearing

in Exercises 20 and 21 above in R^0'1^.

23. For two nets {xa} and {y$} of an ordered vector space show the follow-

ing.

(a) If xa | x and y$ T V-, ^ n e n xa + ys T ^ + V- ^xa T Ax for all A 0,

and Xxa J, Ax for all A 0.

(b) If xa t x, 2/5 T, and xa + y5 | ^5 t n e n 2/5 T ^ - x .

24. Show that for two vectors x and y in an ordered vector space their infimum

x Ay exists if and only if their supremum x V y exists—in which case we

have the identity xA y + xV y — x-hy. [Hint: Use Exercise 18 above. ]

An ordered vector space for which every increasing bounded from above net has a supremum

is called Dedekind (or order) complete. It is not difficult to see that, in this terminology, our

conclusion can be stated as follows: The ordered vector space Rfi is Dedekind complete.