Approximation of Lipschitz functions by Δ-convex functions in banach spaces

a Xi :ma h/9702212 1 [ma h.FA] 13 Feb 1997 APPROXIMATION OF LIPSCHITZ FUNCTIONS BY ∆-CONVEX FUNCTIONS IN BANACH SPACES Manuel Cepedello Boiso Feb ua y 13, 1997 Abs ac . In his pape we gi e some esul s abou he app oxima ion o a Lipschi z unc ion on a Banach space by means o ∆-con ex unc ions. In pa icula , we p o e ha he densi y o ∆-con ex unc ions in he se o Lipschi z unc ions o he opology o uni o m con e gence on bounded se s cha ac e izes he supe e lexi i y o he Banach space. We also show ha Lipschi z unc ions on supe e lexi e Banach spaces a e uni o m limi s on he whole space o ∆-con ex unc ions. 0. In oduc ion and No a ions A unc ion de ined on a Banach space Xis called ∆-con ex i i can be exp essed as a di e ence o con inuous con ex unc ions o , equi alen ly, i i belongs o he linea span o he con inuous con ex unc ions on X. The pu pose o his pape is o gi e some necessa y and su icien condi ions o he app oxima ion o Lipschi z unc ions by ∆-con ex unc ions. The ini ial mo i a ion o ou wo k comes om wo ecen a icles o R. De ille, V. Fon and P. H´ajek ([DFH1] and [DFH2]). A consequence o hei esul s is ha , unde ce ain condi ions on he Banach space X, any con ex unc ion on Xwhich is bounded on bounded se s can be app oxima ed by smoo h con ex unc ions. I is he e o e na u al o conside he class o Banach spaces o which he ∆-con ex unc ions a e dense in he class o Lipschi z unc ions in o de o ex end his p ope y o smoo h app oxima ions. Ou main esul is he ollowing cha ac e iza ion o supe e lexi i y. Theo em 0. Le Xbe a Banach space. Then Xis supe e lexi e i and only i e e y Lipschi z unc ion on Xcan be app oxima ed uni o mly on bounded se s by di e ences o con ex unc ions on Xwhich a e bounded on bounded se s. A consequence o Theo em 0 is ha he abo e men ioned app oach does no p o ide any new esul , because i wo ks only o supe e lexi e spaces ( o which he smoo h app oxima ion p ope y is known using he exis ence o pa i ions o uni y, see Ch. VIII o [DGZ]; o he analy ic app oxima ion case, see [K]). Fo he supe e lexi e case, we gi e explici o mulas o he ∆-con ex app oxi- ma ion o a Lipschi z unc ion. These o mulas, which a e simple han hose om 1991 Ma hema ics Subjec Classi ica ion. P ima y 46B20; Seconda y 46B10. Key wo ds and ph ases. Con ex unc ions, supe e lexi i y in Banach spaces. The au ho was suppo ed by a FPU G an o he Spanish Minis e io de Educaci´on y Ciencia. Typese by A M S-T EX [S ], also p o ide uni o m con e gence on he whole space X. Speci ically, he de- g ee o con e gence gi en by ou o mulas elies di ec ly on he o undi y o he equi alen no m ha i is used. Simila ideas in his di ec ion can be ound in [A] and [PVZ]. We hank P. H´ajek o b inging o ou a en ion he link be ween ou wo k and he dis o ion heo em (see [OS] o de ini ions and de ails). Ou esul s p o ide simple o mulas o deducing, on minimal supe e lexi e Banach spaces (such as ℓp, 1< p < ∞), he exis ence o a con ex unc ion which is no oscilla ion s able om he exis ence o a Lipschi z unc ion which is no oscilla ion s able. Le us ix some no a ion used in his pape . Fo a eal Banach space X, we deno e an equi alen no m on Xby k · k and by BXi s closed uni ball unde his no m. By con ex unc ion we will always mean con inuous con ex unc ion. We will conside wo undamen al opologies on he se o con inuous unc ions de ined on X:τκ( espec i ely τb) is he opology o uni o m con e gence on compac se s o X( esp. uni o m con e gence on bounded se s o X). The modulus o con exi y o he no m k · k de ined by δk·k(ε) = in 1−    x+y 2    :x, y ∈BX;kx−yk ≥ ε(0 < ε < 2) is called o powe ype p(p≥2) i δk·k(ε)≥Kεp, o some K > 0. The concep o dyadic ee will play an impo an ole in he second pa o his wo k. Ou ees a e geome ic ees con ained in Xand de ined as ollows. The symbol αdeno e a mul i-index α= (α1 a α2 a...a αn)∈ {−1,1}<Nand |α|:= n. Fo n∈N, a dyadic (n, θ)- ee Tin Xis a se o he o m xα∈X:α∈ {−1,1}<n sa is ying he ollowing wo condi ions: (1) xα=1 2xα a 1+1 2xα a -1, o all |α|< n. (2) kxα−xα′k ≥ θ > 0, o α6=α′. The poin x∅will be called he oo o he ee T. 1. The posi i e esul s Theo em 1. Le (X, k · k)be a Banach space. The no m k · k is locally uni o mly con ex ( espec i ely uni o mly con ex) i and only i he ollowing p ope y holds: o e e y Lipschi z unc ion on X, he sequence o unc ions ( n)n∈Nde ined by he o mula n(x) := in y∈Xn (y) + n2kxk2+ 2kyk2− kx+yk2o(n∈N, x ∈X) is τκ-con e ging ( esp. τb-con e ging) o . Rema k. Fo any unc ion on Xand n∈N, he unc ion nde ined as abo e is a ∆-con ex unc ion. This ollows immedia ely om he decomposi ion n=cn−dn, wi h cn(x) := 2nkxk2and dn(x) := sup y∈Xnkx+yk2−2nkyk2− (y)(x∈X). The unc ions cnand dna e clea ly con ex. 2 P oo o he Theo em 1. Le us see i s ha he p ope y is necessa y. So, le be a Lipschi z unc ion on X. We ha e o show ha he p e iously de ined sequence ( n)n∈N(τKo τb)- con e ges o i he no m k · k sa is ies he co esponding o undi y condi ion. We begin wi h he ollowing gene al esul : Fac . Fo any poin x∈X, ( n(x))n∈Nis an inc easing sequence bounded abo e by (x). This Fac ollows immedia ely om aking y=xin he n’s in imum o mula and om he inequali y 2kxk2+2kyk2−kx+yk2≥2kxk2+2kyk2−kxk+kyk2=kxk − kyk2≥0.(1) Since o K∈Nwe ha e ha (K·n)=K Kn, he p e ious Fac allows us o suppose wi hou loss o gene ali y ha he Lipschi z cons an o is less han 1 (i.e. | (x)− (y)| ≤ kx−yk o all x, y ∈X). We need o s udy o he in imum o mula de ining na a poin x∈X. Thus, conside any poin y o which we ha e (y) + n2kxk2+ 2kyk2− kx+yk2≤ (x).(2) As is 1-Lipschi z, we deduce om (1) and (2) ha nkxk − kyk2≤n2kxk2+ 2kyk2− kx+yk2≤ (x)− (y)≤ kx−yk.(3) This las condi ion gi es a ela ion be ween he no ms o xand y. Ac ually, kyk can be con olled by kxkin he ollowing way. Suppose ha kyk ≥ 1 + kxk, hen by (3) we ge ha 1≤kxk − kyk≤kxk − kyk2≤1 nkx−yk ≤ 1 nkxk+1 nkyk.(4) And we conclude o n≥3 ha kyk ≤ n+1 n−1kxk ≤ 2kxk.The e o e, we see ha i y sa is ies (2) and n≥3 hen kyk ≤ 2(1 + kxk).(5) Then, o n≥3 we deduce ha n(x) = in kyk≤2(1+kxk) (y) + 2nkxk2+ 2nkyk2−nkx+yk2.(6) In pa icula , he boundedness on bounded se s o n(and, consequen ly, o dn= cn− n) ollows immedia ely. Mo eo e , he uppe bound on kykgi en by (5) oge he wi h he condi ion (3) gi es 0≤2kxk2+ 2kyk2− kx+yk2≤1 nkx−yk ≤ 1 nkxk+1 nkyk ≤ 3 n(1 + kxk).(7) The condi ion (7) gi es he c ucial s ep o he p oo . In ac , i he no m k · k e i ies one o he o undi y p ope ies s a ed in heo em hen by (7) ncan be 3 chosen big enough o necessa ily en o ce y o be close o x. The e o e, n(x) mus be close o (x). Le us jus i y his asse ion. Suppose ha he sequence o unc ions nis no compac ly con e ging o . Since he sequence o ∆-con ex unc ions ( n)nis inc easing and is con inuous, Dini’s heo em ells us ha τκ-con e gence o n o is equi alen o he poin wise con e gence. Then, he e exis s a poin x0∈Xsuch ha ( n(x0))ndoes no con e ge o (x0). As ( n(x0))nis inc easing, he e exis s some ε0>0 such ha o any n∈Nwe ha e n(x0) + ε0< (x0). By de ini ion o n, we can ind a sequence o (yn)nin Xso ha n(x0) + ε0≤ (yn) + 2nkx0k2+ 2nkynk2−nkx0+ynk2+ε0≤ (x0).(8) Since is 1-Lipschi z, we ge om (8) ha kx0−ynk ≥ (x0)− (yn)≥ε0>0.(9) Since yn e i ies he condi ion (8), i ollows om (7) ha we ha e 0≤2kx0k2+ 2kynk2− kx0+ynk2≤3 n(1 + kx0k)−−−→ n→∞ 0.(10) Bu (9) and (10) show ha he no m k · k can no be locally uni o mly con ex a x0(c .Ch. II. P op. 1.2 o [DGZ]). Tha p o es he compac con e gence o n o i he no m k · k is locally uni o mly con ex. A simple p oo o he uni o m con e gence o he sequence nin he uni o mly con ex case ollows he same lines. We will gi e la e a di ec quan i a i e app oach (see he p oo o Theo em 3). I he sequence ndoes no con e ge uni o mly on a bounded se o X, he e is an ε0>0 and a bounded sequence (xn)nso ha n(xn) + ε0< (xn), n∈N. Then, o each xn(n∈N) we can choose yn e i ying (yn) + 2nkxnk2+ 2nkynk2−nkxn+ynk2+ε0≤ (xn).(11) Simila easonings as be o e imply om (11) ha he nex wo s a emen s a e ul iled: kxn−ynk ≥ (xn)− (yn)≥ε0>0 (12) 0≤2kxnk2+ 2kynk2− kxn+ynk2≤3 n(1 + kxnk)−−−→ n→∞ 0 (13) We ha e limn→∞ 3 n(1+kxnk) = 0 since he sequence (xn)nis bounded. Mo eo e , (5) implies ha he sequence (yn) is also bounded. The e o e, (12) and (13) show ha he no m k · k canno be uni o mly con ex (c . Ch. IV Lemma 1.5 [DGZ]). The necessi y o he p ope y is p o ed. Con e sely, he p ope y s a ed in he heo em is su icien . We i s show ha he p ope y o compac con e gence implies he ollowing claim. 4 Claim 1.1. Fo any sequence (xn)n∈N⊂Xand any poin x0∈X, he condi ion limn→∞ 2kx0k2+ 2kxnk2− kx0+xnk2= 0 implies dx0,(xn)n= 0. Since he Claim 1.1 is also alid o any subsequence o he gi en sequence (xn)n, we can s eng hen he conclusion o he claim o limn→∞ kxn−x0k= 0 and he locally con exi y o he no m k · k is e i ied (c .Ch. II. P op. 1.2 o [DGZ]). P oo o he Claim 1.1. Conside he Lipschi z unc ion g(x) = dx, (xn)n,x∈X. Fo x0∈Xand ε > 0, he poin wise con e gence o he sequence (gn)n o ga x0 ells us ha he e exis s Nε∈Nso ha dx0,(xn)n=g(x0)≤in y∈Xng(y) + Nε2kx0k2+ 2kyk2− kx0+yk2o+ε. (14) Using g(xn) = 0, we can e alua e he inequali y (14) a y=xn(n≥Nε) and deduce ha dx0,(xn)n≤Nεlim n→∞ 2kx0k2+ 2kxnk2− kx0+xnk2+ε=ε. The e o e, dx0,(xn)n= 0 and he claim is p o ed. I he p ope y o uni o m con e gence on bounded se s o Xholds, he nex claim, analogous o he p e ious one, also does. Claim 1.2. Le (xn)n∈Nand (yn)n∈Nbe wo bounded sequences in Xsuch ha limn→∞ 2kxnk2+ 2kynk2− kxn+ynk2= 0. Then limm→∞ dym,(xn)n= 0. P oo o he Claim 1.2. As be o e, conside he Lipschi z unc ion d·,(xn)nand ε > 0. This ime, he p ope y o uni o m con e gence gi es a posi i e in ege Nε so ha o all m∈N he nex inequali y is sa is ied. dym,(xn)n≤in z∈Xndz, (xn)n+Nε2kzk2+ 2kymk2− kz+ymk2o+ε. (15) Taking z=xmin he in imum o (15) we ob ain dym,(xn)n≤Nε2kxmk2+ 2kymk2− kxm+ymk2+ε≤2ε, o mla ge enough, and he claim is p o ed. In o de o inish wi h he p oo , we shall show ha he alidi y o Claim 1.2 implies ha he no m k · k is uni o mly con ex. I no , (c . Ch. IV Lemma 1.5 [DGZ]) he e exis s wo bounded sequences (xn)n∈Nand (yn)n∈Nin Xsa is ying ha lim n→∞ 2kxnk2+2kynk2−kxn+ynk2= 0 and kxn−ynk ≥ 1 ( o all n∈N). (16) Fi s , he no m k · k is locally uni o mly con ex (since Claim 1.1 clea ly holds). I ollows ha he sequence (xn)nhas no no m clus e poin . Indeed, i o some x0∈X he e exis s (xnk)k−→ kx0, hen lim k→∞ 2kx0k2+ 2kynkk2− kx0+ynkk2= lim k→∞ 2kx0k2+ 2kxnkk2− kx0+xnkk2= 0 and he e o e (ynk)k−→ kx0. A con adic ion wi h he ac kxn−ynk ≥ 1, o all n∈N, o (16). Hence, passing o a subsequence, we can suppose ha o some 1 > α > 0 we ha e ha kxn−xmk ≥ α, o all n6=m. Now, we need he ollowing echnical lemma, whose p oo will be gi en la e . 5 Lemma 1.3. Le be (λn)n∈N⊂[0,1] and (xn)n∈N,(yn)n∈N wo bounded sequences in Xso ha 2kxnk2+ 2kynk2− kxn+ynk2−−−→ n→∞ 0. Then he sequence o con ex combina ions zn=λnxn+ (1 −λn)yn(n∈N) sa is ies ha 2kxnk2+ 2kznk2− kxn+znk2−−−→ n→∞ 0. Fo any n∈N, ake 0 ≤λn≤1 such ha o zn:= λnxn+ (1 −λn)ynwe ha e kxn−znk=α 2. Then, he Lemma 1.3 and he Claim 1.2 used join ly imply ha limm→∞ d(zm,(xn)) = 0. Bu , kxn−znk=α 2>0 and also o n6=m kxn−zmk ≥ kxn−xmk − kxm−zmk ≥ α−α 2=α 2>0, a con adic ion.  P oo o he Lemma 1.3. By he inequali y 0≤kxnk − kynk2≤2kxnk2+ 2kynk2− kxn+ynk2−−−→ n→∞ 0,(17) we ha e ha lim n→∞ kxnk − kynk= 0. Since he sequences (xn)nand (yn)na e bounded we also ha e ha lim n→∞ kxnk2− kynk2= 0.(18) Bu hen we deduce om (17) and (18) ha lim n→∞ kxnk2−   xn+yn 2   2= lim n→∞ kynk2−   xn+yn 2   2= 0.(19) On he o he hand, using he con exi y o he unc ion k · k2we ge he ollowing gene al lowe es ima e o 0 ≤λ≤1 kλx+(1−λ)yk2≥    x+y 2    2 −1−2λmax (    x+y 2    2 −kxk2 ,    x+y 2    2 −kyk2). Pu ing (17),(18),(19) and he las inequali y oge he , we ob ain ha 0≤2kxnk2+ 2kznk2− kxn+znk2 = 2kxnk2+ 2kλnxn+ (1 −λn)ynk2− k(1 + λn)xn+ (1 −λn)ynk2 ≤2kxnk2+ 2λnkxnk2+ 2(1 −λn)kynk2−4    1 + λn 2xn+1−λn 2yn    2 ≤2kxnk2+ 2kynk2− kxn+ynk2+λnkxnk2− kynk2 + 4λnmax (    xn+yn 2    2 − kxnk2 ,    xn+yn 2    2 − kynk2)−−−→ n→∞ 0. The lemma is p o ed and his concludes he p oo o Theo em 1. Rema k. The in imum o mula used o he de ini ion o ( n) in he Theo em 1 is closely ela ed o he well-known in -con olu ion o mula o by nk · k2:  (nk · k2)(x) := in y∈X (y) + nkx−yk2(n∈N,x∈X).(20) 6 In ac , hese wo in imum o mulas a e iden ical i he no m k · k is a Hilbe ian no m, because o he pa allelog am iden i y. Howe e , o a non-Hilbe ian no m k · k he unc ions gi en by he in -con olu ion o mula can no be exp essed in gene al as ∆-con ex unc ions. As a co olla y o he abo e ema k, we ha e ha he o mula o Theo em 1 con e ges uni o mly on X o case o a Hilbe ian no m k · k (since i is well-known ha he in -con olu ion o mula o (20) con e ges uni o mly on he whole space X, see [LL]). The ques ion ha na u ally a ises is whe he his emains ue o no o a gene al uni o mly con ex no m. The answe is gi en in he ollowing p oposi ion. P oposi ion 2. Le (X, k · k)be a Banach space. I o e e y Lipschi z unc ion on X he sequence o unc ions n(x) := in y∈Xn (y) + n2kxk2+ 2kyk2− kx+yk2o con e ges o uni o mly on X, hen he modulus o con exi y o he no m k · k is o powe ype 2. P oo o he P oposi ion 2. This uni o m con e gence p ope y has he ollowing consequence analogous o Claim 1.1 and Claim 1.2 and whose p oo is iden ical o ha o Claim 1.2. Claim 2.1. I (xn)n∈Nand (yn)n∈Na e wo sequences (no necessa ily bounded) in Xsa is ying ha limn→∞ 2kxnk2+ 2kynk2− kxn+ynk2= 0, hen hey also e i y ha he limm→∞ dym,(xn)n= 0. By Claim 1.2, any no m which sa is ies he conclusion o Claim 2.1 is uni o mly con ex. In ac , Claim 2.1 insu es ha he modulus o con exi y o he no m k · k is o powe ype 2. I no (see [H]), he e exis s wo sequences (xn)n∈Nand (yn)n∈N such ha 2kxnk2+ 2kynk2<kxn+ynk2+1 nkxn−ynk2xn6=yn,n∈N.(21) I we ake un:= xn kxn−ynkand n:= yn kxn−ynkin (21) we ob ain ha 0≤2kunk2+ 2k nk2− kun+ nk2<1 n−−−→ n→∞ 0,(22) kun− nk= 1.(23) Since he no m k·k is uni o mly con ex and condi ions (22) and (23) holds, he sequences (un)nand ( n)ncan no be bounded. No ice ha we ha e also om (22) ha limn→∞ kunk − k nk= 0. Thus, passing o a subsequence we can suppose ha sup{kunk,k nk} + 1 <min{kun+1k,k n+1k} o all n∈N =⇒ kun− mk ≥ kun|| − k mk≥1 (n6=m). (24) Assembling (23) and (24) we deduce ha d m,(un)n= 1 (m∈N). Bu Claim 2.1 and (22) imply ha limmd m,(un)n= 0 which is a con adic ion.  Theo em 3 below p o ides a con e se o P oposi ion 2 and an explici o mula o uni o m app oxima ion o Lipschi z unc ions by ∆-con ex unc ions on supe - e lexi e Banach spaces. The alidi y o his o mula depends upon he exis ence in he supe e lexi e space o an equi alen no m which is enough o und. 7 Theo em 3. Le (X, k · k)be a Banach space whose no m k · k has i s modulus o con exi y o powe ype p(p≥2). Then o e e y Lipschi z unc ion on X he ollowing sequence o ∆-con ex unc ions p n(x) = in y∈Xn (y) + n2p−1kxkp+ 2p−1kykp− kx+ykpo(n∈N, x ∈X) con e ges o uni o mly on X. P oo o he Theo em 3. We essen ially need he ollowing lemma o p o e he esul . Lemma 3.1. I he modulus o con exi y o he no m k·k is o powe ype p(p≥2) hen he e exis s a posi i e cons an Ck·k ≤1such ha o e e y pai x, y ∈X he ollowing inequali y holds: Ck·kkx−ykp≤2p−1kxkp+ 2p−1kykp− kx+ykp. P oo o he Lemma 3.1. Unde he assump ion o he lemma (see [H]), he e exis s a posi i e cons an C′ k·k ≤2 such ha o e e y pai u, ∈Xwe ha e ha ku+ kp+ku− kp≥2kukp+C′ k·kk kp.(25) The lemma ollows om he change o a iables x=u+ 2and y=u− 2in (25). Gi en a Lipschi z unc ion on X, we use he p e ious Lemma 3.1 and he ac ha p n(x)≤ (x) ( o all x∈Xand n∈N) o ob ain he ollowing chain o inequali ies:  (nCk·kk · kp)≤ p n≤ (n∈N).(26) Since p≥2, he p e ious in -con olu ion o mula in (26) con e ges o uni o mly on X(see [LL]). The e o e, he same is ue o ( p n)n. Fo ins ance, i is a 1-Lipschi z unc ion we ge he ollowing es ima e k − p nk∞≤ k − (nCk·kk · kp)k∞≤1 nCk·k 1 p−1 . The i s consequence ha s ems om he Theo em 1 is ano he p oo o a esul due o G.A. Edga ([E]). Co olla y 4. Le XBanach space wi h an equi alen locally uni o mly con ex no m. Then he σ- ields o Bo el se s o he no m and weak opologies a e he same. P oo o he Co olla y 4. The i s pa o he Theo em 1 a i ms ha he exis ence o a locally uni o mly con ex no m on Ximplies he ollowing: ∀ :X→RLipschi z ∃(cn)n,(dn)n⊂Con (X) such ha =τκ- lim n→∞(cn−dn). Le us check ha his p ope y implies he equi alence o he wo Bo el amilies, Bo (X, k · k) and Bo (X, w). Ob iously, Bo (X, w)⊆Bo (X, k · k). To see he o he inclusion, ake Fak · k-closed se o X. Then conside he Lipschi z unc ion (·) = dis (·, F). No e ha he p e ious p ope y implies ha is he poin wise limi o a sequence o w-Bo el unc ions (because e e y con inuous con ex unc ion is w-lowe -semicon inuous). The e o e, is w-Bo el and so Fis w-Bo el.  Now, we p oceed o s a e wo applica ions o Theo em 1 and Theo em 3 o he s udy o supe e lexi e Banach spaces. Ac ually, we will show in he nex sec ion ha bo h o hem cha ac e ize supe e lexi i y. 8 Co olla y 5. Le Xbe a supe e lexi e Banach space. Deno e by Con (X) he se o con inuous con ex unc ions on X,Con b(X) he subse o Con (X)consis ing o he unc ions which a e bounded on bounded se s and UCb(X) he class o unc ions on Xwhich a e uni o mly con inuous on bounded se s o X. Then spanτbCon b(X)=UCb(X). P oo o he Co olla y 5. F om he P op. 1.6 o [Ph] ollows immedia ely ha Con b(X) = Con (X)∩ UCb(X)⊆ UCb(X). Le us show he τb-densi y o he ∆-con ex unc ions in he se UCb(X). No ice ha he se o Lipschi z unc ions on Xis dense in UCb(X) unde he opology τb. This ac can be p o ed using again he in -con olu ion o mula. Mo e p ecisely, gi en ∈ UCb(X) and bounded on X, (nk · k)(x) is a sequence o Lipschi z unc ions τb-con e ging o . Since he bounded unc ions o UCb(X) a e clea ly τb-dense in UCb(X), he densi y o he Lipschi z unc ions is he e o e deduced. Thus, he co olla y holds i we show ha he con ex unc ions {cn, dn}n∈Nob- ained in he p oo o Theo em 1 a e in Con b(X). Clea ly, cn(·) = 2nk · k2∈ Con b(X) and, as we ema ked du ing he p oo o his heo em, dn=cn− nis also bounded on bounded se s o X. Co olla y 6. Le Xbe a supe e lexi e Banach space. Wi h he same no a ions o he p e ious co olla y, one has spanτuCon b(X)⊃ UC(X) (whe e τuis he opology o uni o m con e gence on X). P oo o he Co olla y 6. As we ema ked du ing he p oo o he p e ious Co ol- la y 5, he esul is p o ed i we show he τu-densi y o he subse o ∆-con ex unc ions Con b(X) in he se o uni o mly con inuous unc ions UC(X). Bu his ollows om Theo em 3 and Pisie ’s eno ming heo em ([P]) ha gi es an equi alen no m wi h modulus o con exi y o powe ype p( o some p≥2) on e e y supe e lexi e Banach space. Rema k. Fo a simple and mo e geome ical p oo o Pisie ’s heo em we e e o [L]. 2. The nega i e esul s In his pa , we will show ha he o undi y condi ions on he no m needed in Theo em 1 can no be d opped. Fo ins ance, e en he poin wise con e gence ails o some Banach spaces, as he ollowing coun e -example shows. Example 7. The e exis s a Lipschi z unc ion on ℓ∞which can no be a poin wise limi o a sequence o ∆-con ex unc ions. In he a icle [T], M. Talag and p o ed ha Bo (ℓ∞, w)&Bo (ℓ∞,k · k∞). Tak- ing a k · k∞-closed, non w-Bo el se B he unc ion d(·, B) is a Lipschi z unc ion which can no be he poin wise limi o any sequence o ∆-con ex unc ions (since ∆-con ex unc ions a e w-Bo el). On he o he hand, he τb-densi y p ope y o he span o Con b(X) in UCbis a cha ac e iza ion o supe e lexi i y. This conclusion comes om he ollowing heo em. 9