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 FACULTY WORKING 
 PAPER NO. 1202 
 
 ob 
 
 
 On Sufficient Conditions for the Sum of Two 
 Weak * Closed Convex Sets to be Weak * Closed 
 
 M. AH Khan 
 Rajiv Vohra 
 
 College of Commerce and Business Administration 
 Bureau of Economic and Business Research 
 University of Illinois, Urbana-Champaign 
 
BEBR 
 
 FACULTY WORKING PAPER NO. 1202 
 College of Commerce and Business Administration 
 University of Illinois at Urbana-Champaign 
 October, 1985 
 
 On Sufficient Conditions for the Sum of Two 
 Weak * Closed Convex Sets to be Weak * Closed 
 
 M. Ali Khan, Professor 
 Department of Economics 
 
 Rajiv Vohra 
 Brown University 
 
 This research was supported, in part, by a NSF grant to Brown University, 
 
Digitized by the Internet Archive 
 
 in 2011 with funding from 
 
 University of Illinois Urbana-Champaign 
 
 http://www.archive.org/details/onsufficientcond1202khan 
 
On Sufficient Conditions for the Sum of Two Weak * Closed 
 Convex Sets to be Weak * Closed 
 
 by 
 
 M. Ali Khan and Rajiv Vohra 
 
 September 1985 
 
 Abstract . We give a sufficient condition for the sum of two weak * 
 closed convex subsets of a strictly hypercomplete locally convex 
 Hausdorff topological vector space to be weak * closed. Our suffi- 
 cient condition relates to the mutual position of the two sets and our 
 result may be seen as a simple consequence of the Alaoglu-Bourbaki 
 Theorem. 
 
 t 
 
 This research was supported, in part, by a NSF grant to Brown University. 
 
1. 
 
 The question as to when the sum of two closed sets is closed arises 
 naturally in applied mathematics and is of interest in its own right. 
 For subsets of an infinite dimensional space, answers to this question 
 have been given by Choquet [1] , Ky Fan [3] and Dieudonne [2]. Ky Fan, 
 in particular, has shown that the sum of two weakly closed convex sub- 
 sets of a locally convex topological vector space is weakly closed under 
 a condition that takes account of the mutual position of the two sets. 
 In this note, we offer another sufficient condition of this genre for 
 subsets in the dual of a locally convex topological vector space. We 
 also give simple examples which bring out the difference between our 
 sufficient condition and those of the others. It is worth mentioning 
 that our result is directly inspired by a problem in mathematical eco- 
 nomics, see [4], 
 
 2. 
 
 Let (E,x) be a real, Hausdorff locally convex vector space and E' 
 its topological dual, i.e., the space of all x-continuous linear func- 
 tions on E. a(E',E) denotes the weak * topology on E'. Following [5, 
 Definition 12-3-7], we say that E is strictly hypercomplete if for any 
 S C E', S/^VU is weak * compact implies S is weak * closed, where U° 
 is the polar of any neighborhood of zero in E. 
 
 We can now state 
 
 Theorem . Let X and Y be two q(E ' ,E)-closed, convex subsets of E' such 
 that Xf~\(-Y+U°) is x-equicontinuous for any neighborhood U of zero in 
 E. If E is strictly hypercomplete, then X+Y is q(E ' ,E)-closed . 
 
-2- 
 
 Proof . Let W = X+Y and for any neighborhood U of zero in E, let W = 
 (X+Y) n U°. Given Alaoglu-Bourbaki theorem, the strict hypercomplete- 
 ness of E and the convexity of X and Y, we need only show that W is 
 a(E' ,E)-closed. Towards this end, pick any net {w } from W and such 
 that the a(E',E) limit of {w } is w. For each w , there exist x e X, 
 y e Y such that x + y = w . Under the hypothesis of the theorem, 
 this implies that {x } lie in a x-equicontinuous set. By a second 
 application of Alaoglu-Bourbaki, we can assert the existence of a sub- 
 net {x } which tends in a(E ? ,E) to a limit, say x. Since X is a(E',E)- 
 
 ^ . V0 vo vd vo , , 
 
 closed, x e X. Furthermore, since y=w - x , y also tends in 
 
 a(E',E) to a limit, say y. Since Y is a(E',E) - closed, y e Y. 
 
 Certainly y = w - x which implies that w e W and the proof is 
 
 finished. 
 
 Corollary . The theorem is true if E is assumed, in addition, to be 
 quasibarrelled and the condition XO(-Y+U°) is x-equicontinuous is 
 weakened to the requirement that it be B (E ' ,E )-bounded, where 8 denotes 
 the strong topology . 
 
 Proof . Given the definition of quasibarrelled [5; Definition 10-1-7 
 and Theorem 10-1-11], an obvious modification in the proof of the 
 theorem yields the corollary. 
 
 3. 
 We now present six examples to relate our result to previous work. 
 Our first two examples are taken from Ky Fan. 
 
 2 2 
 
 Example 1 . In the Euclidean plane R , let X = ((a 1 ,a~) e R : a.. > 0, 
 
 o 
 a,a ? J> l} and Y = {(a.,aJ e R : -°° < a, < » , a., = 0} . In the context 
 
-3- 
 
 of our theorem X+Y cannot be shown to be closed since X/*\(Y+S) or Y O 
 
 (-X+S) is not a compact set, where S denotes the closed unit ball in 
 
 2 
 R . Note that Ky Fan's reason for X+Y not being closed is that the 
 
 interior of X° does not intersect Y°. 
 
 Example 2 . Let X be as in Example 1 but Y = {(a., a,,) e R : a.. + a~ = 0} 
 X+Y is closed since X O (Y+kS) is compact for any positive real number k. 
 This example also satisfies the sufficiency conditions of the results 
 of Ky Fan and Dieudonne but not for that of Choquet. 
 
 2 
 Example 3 . Let X = {(a, ,a 2 ) e R : -°° < a, < °°, a 2 = 0} and Y = 
 
 •y 
 
 {(a 1 ,a_) e R : -« < a~ < °°, a.. = 0} . Since X° and Y° have empty 
 
 2 
 interiors in R , we cannot apply Ky Fan's Theorem 2. Since X and Y are 
 
 not subsets of a convex set that contains no straight line, we cannot 
 
 apply Choquet 's result either. However (X+Y) is closed as a consequence 
 
 of our theorem, since X^\(Y+S) is compact. Note that this example 
 
 also satisfies the sufficiency conditions of Dieudonne 's result. 
 
 Example 4 . In the space % , let X and Y be the positive cone {a e I : 
 a. > 0}. X° = Y° = {a e I : a. <_ 0}. Since the positive cone in I 
 has an empty norm interior and the norm topology is identical to the 
 Mackey topology x(£_,4 ), we cannot appeal to Ky Fan's theorem to 
 assert the closedness of X+Y. Dieudonne 's theorem does not apply since 
 X and Y are not locally compact. Since I is not weakly complete, 
 neither are X and Y and thus Choquet 's result does not apply. However, 
 X+Y is closed as a consequence of our theorem since XC\(-Y+\J°) is 
 
 00 
 
 o(£ >^i) compact for any zero neighborhood U in I . 
 
-4- 
 
 Next, we present an example that does not satisfy any of the suffi- 
 cient conditions required by the results in [1], [2] or [3] but does 
 satisfy those required for our theorem. 
 
 Example 5 . In the Hilbert space £«, let X = {a e L: a. = 0, i even; 
 -» < a < °°, i odd} and Y = {a e %'. a. = 0, i odd; -°° < a. < », i even} 
 Certainly X and Y are not subsets of a convex set that contains no 
 straight line as required in [1] and X,Y are not locally compact in the 
 weak topology, a(£~,£~), as required in [2]. Finally since X° = Y and 
 Y° = X, the Mackey topology is identical to the norm topology, and the 
 norm interior of X and Y is empty, the sufficiency condition for the 
 result in [3] is violated. However X+Y is a(j£ 9 ,£ 9 )-closed since X and 
 Y are a ( 2, «,£'?) -closed an ^ ^O (-X+kU) is o(l~, 2,„)-compact for any real 
 positive number k and U the unit ball in l~. This implies that Y (~\ 
 (-X+kU) is norm equicontinuous in £_. 
 
 Finally, we present an example which does not satisfy any of the 
 sufficient conditions required by the results in [1], [2], [3] or by 
 those in our theorem but the sets are nevertheless closed. 
 
 2 
 Example 6 . Let X = {(a ,a„) e R : a >_ 0, a~ _> 0} and Y = {(a ,a») e 
 
 2 
 R : -°° < a n < oo, a„ _> 0} . Certainly, Y is not a subset of a set that 
 
 contains no straight line as required in [1]. Also, the intersection 
 
 of the asymptotic cones of X and Y is not {0} as required in [2]. The 
 
 interior of X° does not intersect Y° as required in [3], Finally, it 
 
 is clear that X Pi (-Y+S) is not relatively compact. Nevertheless a 
 
 simple computation yields that X+Y is closed. 
 
-5- 
 
 Ref erences 
 
 [1] G. CHOQUET, Ensembles et cones convexes faiblement complets, I, 
 Comptes Rendus Academie Sciences Paris 254 (1962) 1908-1910. 
 
 [2] J. DIEUDONNE, Sur la separation des ensembles convexes, Math. 
 Ann. 163 (1966) 1-3. 
 
 [3] K. FAN, A generalization of the Alaoglu-Bourbaki theorem and its 
 applications, Math. Zeitschr. 88 (1965) 48-60. 
 
 [4] M. ALI KHAN and R. VOHRA, Approximate equilibrium theory with 
 
 infinitely many commodities, Brown University Working Paper , 1985, 
 
 [5] A. WTLANSKY, Modern Methods in Topological Vector Spaces , 1974, 
 McGraw Hill, New York. 
 
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