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.
