Finally, we present an example which does not satisfy and of the sufficient conditions required by the results in [1], [2], [3], [4] or by those in our theorem but the sets are neverthel[r]
Trang 1Arch Math., Vol 48, 328-330 (1987) 0003-889 X/87/4804-0328 $ 2.10/0
9 1987 Birkh/iuser Verlag, Basel
O n sufficient c o n d i t i o n s for the s u m o f t w o w e a k 9 closed
c o n v e x sets to be w e a k 9 closed
By
M ALI KHAN and RMIV VOI-IRA *)
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 F o r subsets of an infinite dimen- sional space, answers to this question have been given by Choquet [1], K y F a n [3], Dieudonn+ [2] and more recently by Jameson [4] K y Fan, in particular, has shown that the sum of two weakly closed convex subsets 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 Despite its simplicity, this result seems to have been overlooked in the literature 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 economies, see [5]
2 Let (E, z) be a real, Hausdorff locally convex vector space and E' its topological dual, i.e., the space of all z-continuous linear functions on E a (E', E) denotes the weak 9 topology on E' Following [6, Definition 12-3-7], we say that E is strictly hypercomplete
if for any S c E', S n U ~ is weak 9 compact implies S is weak 9 closed, where U ~ is the polar of any neighborhood of zero in E
We can now state
Theorem Let X and Y be two a(E', E)-closed, convex subsets of E' such that
X c~ ( - Y + U ~ is z-equicontinuous for any neighborhood U of zero in E I f E is strictly hypercomplete, then X + Y is a (E', E)-closed
P r o o f Let W = X + Y and for any neighborhood U of zero in E, let
W, = (X + Y) c~ U ~ Given Alaoglu-Bourbaki theorem, the strict hypercompleteness 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 v} from W, and such that the a (E', E) limit of {w v} is w F o r each
w v, there exist x v e X, y~ ~ Ysuch that x ~ + y~ = w ~ Under the hypothesis of the theorem, this implies that {x v} lie in a z-equicontinuous set By a second application of Maoglu- Bourbaki, we can assert the existence of a subnet {x ~} which tends in a (E', E) to a limit,
*) This research was supported, in part, by a NSF grant to Brown University
Trang 2Vol 48, 1987 Sum of two weak 9 closed convex sets 329
say x Since X is ~ (E', E)-closed, x ~ X F u r t h e r m o r e , since yVe = wVe _ x~O, y~O also tends
in cr (E', E) to a limit, say y Since Y is tr (E', E)-closed, y ~ Y Certainly y = w - x which
implies t h a t w s W, a n d the p r o o f is finished []
Corollary The theorem is true if E is assumed, in addition, to be quasibarrelled and the condition X • ( - Y + U ~ is z-equicontinuous is weakened to the requirement that it be
fl (E', E)-bounded, where fl denotes the strong topology
P r o o f G i v e n the definition of quasibarrelled [6; Definition 10-I-7 a n d T h e o r e m 10-1-11], an obvious modification in the p r o o f of the t h e o r e m yields the corollary []
3 W e now present six examples to relate o u r result to previous work O u r first two examples are taken from K y F a n
E x a m p l e 1 In the Euclidean plane R 2, let X = { ( a l , a z ) e R 2 : a 1 > O, a l a 2 > 1}
a n d Y = {(a 1, a2)E R2: oO < a 1 < 0% a 2 - 0} I n the context of o u r t h e o r e m X + Y
c a n n o t be shown to be closed since X n (Y + S) or Y n ( - X + S) is n o t a c o m p a c t set, where S denotes the closed unit ball in R 2 N o t e t h a t K y F a n ' s reason for X + Y not being closed is that the interior of X ~ does not intersect y 0
E x a m p 1 e 2 Let X be as in E x a m p l e I but Y = {(al, a2) ~ R2: a 1 + a2 = 0} X + Y
is closed since X n (Y + k S) is c o m p a c t for any positive real n u m b e r k This example also satisfies the sufficiency conditions of the results of K y F a n a n d D i e u d o n n 6 but not for t h a t
of Choquet
E x a m p l e 3 L e t X = { ( a l , a 2 ) ~ R 2 : - ~ < aa <o%a2 = 0 } a n d Y = {(aa,a2)~R2:
- ~ < a2 < o% a 1 = 0} Since X ~ a n d yO have e m p t y interiors in R 2, we c a n n o t a p p l y
K y F a n ' s T h e o r e m 2 Since X a n d Y are n o t subsets of a convex set t h a t contains no straight line, we c a n n o t a p p l y C h o q u e t ' s result either H o w e v e r (X + Y) is closed as a consequence of o u r theorem, since X n (Y + S) is compact N o t e that this example also satisfies the sufficiency conditions of Dieudonn6's result
Next, we present an example that does not satisfy a n y of the sufficient conditions required by the results in [1], [2] or [3] but does satisfy those required for o u r theorem
E x a m p 1 e 4 In the space #~o, let X a n d Y be the positive cone {a ~ f ~ : al > 0}
X 0 = y o = {a ~ d~: a i < 0} Since the positive cone in f l has an e m p t y n o r m interior a n d the n o r m t o p o l o g y is identical to the M a c k e y t o p o l o g y z ( f l , d~o), we c a n n o t a p p e a l to
K y F a n ' s t h e o r e m to assert the closedness o f X + Y Dieudonn6's t h e o r e m does not a p p l y since X a n d Y are not locally compact Since E~ is n o t w e a k l y complete, neither are X
a n d Y a n d thus C h o q u e t ' s result does n o t apply However, X + Y is closed as a conse- quence of o u r t h e o r e m since X n ( - Y + U ~ is a (E ~ f~) c o m p a c t for a n y zero neighbor-
h o o d U in d~ It is also closed as a consequence of P r o p o s i t i o n 5 i n Jameson It is of interest t h a t J a m e s o n proves his p r o p o s i t i o n as a c o r o l l a r y of the K r e i n - S m u l i a n theorem
E x a m p 1 e 5 In the H i l b e r t space f2, let X = {a e E2: ai = 0, i even; - oo < ai < o%
i odd} a n d Y = {a s f z : ai = 0, i odd; - oo < ai < o% i even} Certainly X a n d Y are not
subsets of a convex set that contains no straight line as required in [1] a n d X, Y are n o t
Trang 3330 M ALl KHAN and R VOHRA ARCH MATH locally compact in the weak topology, cr (fz, g~ as required in [2] Finally since X ~ = Y and yO = X, the Mackey topology is identical to the n o r m topology, and the n o r m interior of X and Y is empty, the sufficiency condition for the result in [3] is violated However, X + Y is ff(g~ fz)-closed since X and Y are o'(fz, g~ and
Y n ( - X + k U) is o- (•2, E2)-compact for any real positive number k and U the unit ball
in f2 This implies that Y c~ ( - X + k U) is n o r m equicontinuous in {2 X + Y is also closed as a consequence of Corollary 3 to Theorem 1 in Jameson [4]
Next, we present an example that does not satisfy any of the sufficient conditions required by the results in [1], [2], [3], or [4] but does satisfy those required for our theorem
E x a m p 1 e 6 In the space ~'~, let X = {a e Yo~: ai > 0} and Y be the sum of X and {a~Y~o: Hall < 1} Certainly Y is weak 9 closed Since Y is neither a wedge nor a subspace, none of the results in [4] apply We now leave it to the reader to convince himself that this example does not satisfy the sufficiency conditions in [1], [2] and [3] but does those of our theorem
Finally, we present an example which does not satisfy and of the sufficient conditions required by the results in [1], [2], [3], [4] or by those in our theorem but the sets are nevertheless closed
E x a m p 1 e 7 Let X = {(al, a2) E R2: al > 0, a 2 ~ 0} and
Y = { ( a t , a 2 ) ~ R 2 : o o < a 1 < o % a 2 > 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 o as required in [3] Finally, it is clear that
X c~ ( - Y + S) is not relatively compact Nevertheless a simple computation yields that
X + Y is closed
References
[1] G CHOQUET, Ensembles et cones convexes faiblement complets, I Comptes Rendus Academic Sciences Paris 254, 1908-1910 (1962)
[2] J DIEUDONNI~, Sur la s6paration des ensembles convexes Math Ann 163, 1-3 (1966) [3] K FAN, A generalization of the Alaoglu-Bourbaki theorem and its applications Math Z 88, 48-60 (1965)
[4] G J O JAMESON, The duality of pairs of wedges Proc London Math Soc 24, 531-547 (1972) [5] M ALI KnAN and R VOHRA, Approximate equilibrium theory with infinitely many commodities Brown University Working Paper 1985
[6] A WmANSKV, Modern Methods in Topological Vector Spaces New York 1974
Eingegangen am 20 12 1985 Anschrift der Autoren:
M Ali Khan
University of Illinois at Urbana-Champaign
Department of Economics
1206 South Sixth Street
Champaign, Illinois 61820
USA
R Vohra Brown University Department of Economics Providence, R I 02912 USA