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 x a + 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.
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