AN EXTENSION OF SIMONS INEQUALITY AND APPLICATIONS. Robert DEVILLE and Catherine FINET

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1 2001 vol. XIV, um. 1, ISSN AN EXTENSION OF SIMONS INEQUALITY AND APPLICATIONS Robert DEVILLE ad Catherie FINET Abstract This article is devoted to a extesio of Simos iequality. As a cosequece, havig a poitwise covergig sequece of fuctios, we get criteria of uiform covergece of a associated sequece of fuctios. I Itroductio Simos iequality is a useful tool i Baach space geometry. Simos has observed i [S1] that this iequality allows to prove that if f is a uiformly bouded sequece of real valued cotiuous fuctios o a compact space which coverges poitwise to a cotiuous fuctio g, the there is a sequece of covex combiatios of the f s that coverges uiformly to g. Later, Godefroy [G] foud other applicatios of this iequality see also [FG] ad [GZ]. Ad more recetly, Acosta ad Galá [AG] improved James theorem i the case of smooth Baach spaces. Our mai result is the followig extesio of Simos iequality [S1]. We believe that this extesio may have applicatios i o liear aalysis. Ackowledgemet. The authors thak the referee for the remarks he made o this article. The research preseted i this paper was supported by a FNRS grat ad La Baque Natioale de Belgique. The paper was writte whe the first author visited the Departmet of Mathematics of the Uiversity of Mos-Haiaut Belgium Mathematics Subject Classificatio: 46B20. Servicio de Publicacioes. Uiversidad Complutese. Madrid,

2 II. Mai result Theorem 1. Let B be a set ad C be a o empty subset of a liear ormed space that is stable with respect to takig ifiite covex combiatios. Let f : C B R be a bouded fuctio such that the mappigs x fx, b are covex ad Lipschitz cotiuous, with a Lipschitz costat idepedat of b. Let us also assume that { for every x C there is a b B such that fx, b = sup β B fx, β The if x is a sequece i C, we have if sup x C β B fx, β sup β B lim sup fx, β. I particular, if we take as C a certai subset of l B, the Baach space of all bouded real fuctios o B, we get the classical Simos iequality. Corollary. Simos iequality. Let B be a set ad C be a o empty bouded subset of l B that is stable with respect to takig ifiite covex combiatios. Let us assume that for every x C, there is a b B such that xb = sup xβ β B The if x is a sequece i C, we have if sup x C β B xβ sup β B lim sup x β Let us ow discuss the assumptio C is stable by takig ifiite covex combiatios. This assumptio is clearly satisfied if C is a closed covex subset of a Baach space. O the other had, it is always satisfied by bouded covex subsets of a fiite dimesioal vector space V. This ca be proved by iductio. Ideed, this is clear if the dimesio of the space is equal to 1. We ca assume that 0 C. C satisfies oe of the followig coditios: either C is cotaied i a liear proper subspace of V or C has o empty iterior. I the first case, the statemet follows vol. XIV, um. 1,

3 from our assumptio. If C has o empty iterior, let us assume that the result holds for vector spaces of dimesio. Let C be a bouded covex subset of a vector space V of dimesio + 1. Let us assume that there exist poits x i C ad scalars λ > 0 such that ad that + + λ = 1 λ x / C. By Hah-Baach Theorem, there exists ϕ V such + ϕ λ x = 1 ad ϕx 1 for x C There exists 0 such that ϕx 0 < 1. Ideed, otherwise the iductio hypothesis would ot be satisfied for the covex C {ϕ = 1}. But + + ϕ λ x = λ ϕx < 1 ad this gives us a cotradictio. Lookig at the extesio of Simos iequality we got, it is atural to recall a Mi-Max Theorem see [A], see also [S2] ad to compare both results. Recall that if C is a covex subset of a vector space V, the fiite topology o C is the strogest topology for which, for each ad for each -uple K = y 1, y 2,..., y of elemets i C, the mappigs f K : C + C defied by f K λ 1, λ 2,..., λ = i=1 λ iy i are cotiuous, where C + is the set of all λ 1, λ 2,..., λ R such that λ i 0 for all i ad i=1 λ i = 1. Mi-max theorem. Let B be a compact space ad let C be a covex subset of a vector space V, supplied with the fiite topology. Assume that i for all x C, b fx, b is upper semicotiuous o B, ii for all b B, x fx, b is covex. The there exists b 0 B such that if fx, b 0 = sup if fx, b = sup x C b B x C if c CC,B x C fx, cx vol. XIV, um. 1,

4 where CC, B deotes the space of cotiuous fuctios from C to B. The authors cojecture that this Mi-Max Theorem should be deduced from Theorem 1. Proof of theorem 1. Let us cosider, for x i C, σx = sup fx, b ad let us put, m = if {σx, x C} M = sup {σx, x C}. Sice f is bouded o C B, we have < m M <. Let x be a sequece i C ad put C p = cov {x, p}. We ca assume m > 0. Let 0 < δ < m. Let a p be a sequece such that 0 < a p 1, a p = 1 ad a p δ M a, ad let ɛ be a sequece such that where A = p 1 1 p a p. 0 < ɛ a +1a + a +1 2A +1 δ. Let y 1 C 1 be such that σy 1 if y C 1 σy + ɛ 1. If y 1, y 2,..., y 1 have bee chose, we write z 1 = 1 y i C such that z + a y σ if σ + ɛ, A y C A k=1 b B a k y k ad take Now, put z = p 1 a p y p. Clearly, z C, so, by assumptio, there exists b i B such that, fz, b = σz. Sice a z 1 p y p z = + a y + a p, a p vol. XIV, um. 1,

5 by covexity of f with respect to the first variable, we get : a p y p fz, b f, b + a fy, b + a p f, b. a p Therefore, a fy, b σz σ a p M Hece, by the choice of a, 1 a fy, b σz σ δa. Sice f is Lipschitz cotiuous with respect to the first variable, with Lipschitz costat idepedet of the secod variable, σ is Lipschitz zp cotiuous. Therefore, sice lim A = 1, the lim σ p A p σz p = 0, zp ad so σz = lim p A p σ A p, ad σz σ p [A p 1 σ zp = p zp 1 σ A p 1 A p [ zp A p σ A p zp 1 A p 1 σ ] + a p m. A p 1 ] zp zp 1 Let us put p = σ A p σ A p 1. The followig lemma will lead us to a good estimate of p A p 1 p. We will give the proof of this lemma after the ed of the proof of the theorem. Lemma. We have, for every 2, a δ. It follows from the lemma that p A p 1 p δ p A p 1 a p δ p a p vol. XIV, um. 1,

6 Therefore, σz σ This estimate ad 1 yield fy, b m 2δ. a p m δ a m δ. p As y C, by covexity of f i the first variable, for each there exists k such that f x k, b m 2δ. So, sup lim sup fx, b m 2δ, for all δ > 0. Thus the theorem is proved. b We ow give the proof of the lemma: Proof of the lemma. We first claim that : 2 ɛ 1 ad for > 2, +1 γ 2ɛ, where γ = a+1 a A +1. z2 z1 Ideed, 2 = σ σ. As z 2 A 2 C 1, = y 1, by defiitio A 2 A 1 of y 1, 2 ɛ 1. Let r = a +1 ad y = y + r y +1. Sice y C, by the choice of z a 1 + r it holds σ z A σ z+1 + r z 1 A 1 + r z 1 A 1 + ɛ. We have A 1 + r = A +1 + r, so z +1 z z A+1 A σ σ +1 + r 1 + ɛ. A A +1 + r Ad, by covexity, A +1 + r σ z A A +1 σ z+1 A +1 + r σ This iequality ca be rewritte as follows : ] z+1 z A +1 [σ σ r [σ A +1 A +A +1 + r ɛ. z A A +1 + r ɛ ] σ vol. XIV, um. 1,

7 fially we get that +1 r A +1 this proves the claim. 1 + r ɛ γ 2ɛ A +1 The lemma the follows easily by iductio from the claim ad from the choice of the sequece ɛ. III Applicatios I this sectio, we preset some applicatios of Theorem 1 which caot be deduced from Simos iequality. Recall that the covex hull of a N sequece x is the set of fiite combiatios λ x with λ 0 for all ad N λ = 1. Theorem 2. Let B be a set, X be a Baach space, C be a closed covex subset of X ad f : C B R be a bouded fuctio such that the mappigs x fx, b are covex ad Lipschitz cotiuous, with a Lipschitz costat idepedet of b. Let us assume that for every x C there exists a b B such that fx, b = sup fx, β β B If x is a sequece i C such that for every β B, fx, β 0 ad lim fx, β = 0, the, for all ɛ > 0, there exists x i the covex hull of the sequece x such that sup fx, β ɛ β B vol. XIV, um. 1,

8 Of course, whe f takes values i the positive real umbers, if for every β B, fx, β coverges poitwise to 0, the coclusio of Theorem 2 is that there exists a sequece y of covex combiatios of x such that fy, β coverges to 0 uiformly with respect to β. Proof. Ideed, we have sup lim sup fx, β = 0. Let us deote C the β B closed covex hull of the sequece x. Theorem 1 shows that if sup fx, β 0 x C β B Sice the covex hull of the sequece x is dese i C, the above iequality ad the Lipschitz cotiuity of f with respect to the first variable imply Theorem 2. If we take B = N i the above theorem, we get the followig result. Corollary. Let C be a closed covex subset of a Baach space X ad f be a sequece of covex cotiuous fuctios from C to R, which is uiformly bouded ad uiformly Lipschitz o C. Let us assume that for every x C there exists a 0 N such that f 0 x = sup f x If x is a sequece i C such that for every p N, f p x 0 ad lim f p x = 0; the, for all ɛ > 0, there exists x i the covex hull of the sequece x such that sup f p x ɛ p N Remark. The hypothesis of the covexity of f caot be dropped. Ideed, cosider X = R, f x = if {x + +, 1} ad a sequece x tedig to. O the other had, if you take f x = x + +, you see that the hypothesis of the uiform boudedess of the sequece f also caot be dropped. Let us recall the followig result see [S1, Corollary 10]. Let K be a compact space ad f be a uiformly bouded sequece of cotiuous fuctios o K. If the sequece f coverges poitwise to zero o K the it coverges weakly to zero vol. XIV, um. 1,

9 We ow give a vector-valued extesio of this result. Propositio. Let K be a compact space, X be a Baach space ad f be a uiformly bouded sequece of cotiuous fuctios from K to X. If the sequece f coverges poitwise to 0 o K the there exists a sequece of liear covex combiatios of f which is uiformly coverget o K. Proof. Let us deote C the closed covex hull of the fuctios f i the Baach space CK, X of cotiuous fuctios from K ito X. The fuctio F : C K R defied by F f, x := fx X is bouded, covex ad cotiuous with respect to the first variable, ad, for every f C, there exists x K such that fx X = sup y K fy X. By assumptio, sup x K lim sup This proves the propositio. f x X = 0. Accordig to Theorem 1, if sup fx X 0 f C x K Let us metio that, by a remark of the referee, this result is also a cosequece of Simo s iequality. The subset K B X is a boudary of CK, X. If f coverges poitwise to zero o K, it coverges poitwise o the boudary ad so, it coverges weakly to zero. Refereces [AG] [A] [DGZ] [FG] M. D. Acosta ad M. Ruiz Galá, New characterizatios of the reflexivity i terms of the set of orm attaiig fuctioals. Caad. Math. Bull , J.P. Aubi, Mathematical methods of game ad ecoomic theory. Studies i Mathematics ad its applicatios. Vol. 7. North Hollad, R. Deville, G. Godefroy ad V. Zizler, Smoothess ad reormigs i Baach spaces. Pitma moographs ad Surveys i Pure ad Appl. Math.. Logma Scietific & Techical, M. Fabia ad G. Godefroy, The dual of every Asplud space admits a projective resolutio of idetity. Studia Math , vol. XIV, um. 1,

10 [G] [GZ] G. Godefroy, Boudaries of a covex set ad iterpolatio sets. Math. A , G. Godefroy ad V. Zizler, Roughess properties of orms o o- Asplud spaces. Michiga Math. J , [J] R.C. James, Weakly compact sets. Tras. Amer. Math. Soc , [O] [S1] [S2] E. Oja, A proof of Simos iequality, preprit. S. Simos, A covergece theorem with boudary. Pacific J. Math. 40, 1972, S. Simos, Miimax ad mootoicity. Lecture Notes i Math. Vol. 1693, Spriger Verlag, Robert Deville Uiversité de Bordeaux I Mathématiques Pures de Bordeaux Cours de la Libératio, 351, F Talece Cedex Frace deville@math.u-bordeaux.fr Catherie Fiet Uiversité de Mos-Haiaut Istitut de Mathématique et d Iformatique Le Petagoe Aveue du Champ de Mars, 6 B 7000 Mos Belgique catherie.fiet@umh.ac.be Recibido: 17 de Noviembre de 1999 Revisado: 3 de Julio de vol. XIV, um. 1,