On the range of the derivative of Gâteaux-smooth functions on separable Banach spaces

Transcription

1 O the rage of the derivative of Gâteaux-smooth fuctios o separable Baach spaces Robert Deville Mathematiques Pures de Bordeaux, Uiversité de Bordeaux 351, cours de la libératio, 33400, Talece, Frace deville@math.u-bordeaux.fr Petr Háje 1 Mathematical Istitute, Czech Academy of Sciece Žitá 25, Prague, Czech republic haje@math.cas.cz Abstract : We prove that there exists a Lipschitz fuctio from l 1 ito IR 2 which is Gâteaux-differetiable at every poit ad such that for every x,y l 1, the orm of f (x) f (y) is bigger tha 1. O the other had, for every Lipschitz ad Gâteaux-differetiable fuctio from a arbitrary Baach space X ito IR ad for every ε > 0, there always exists two poits x,y X such that f (x) f (y) is less tha ε. We also costruct, i every ifiite dimesioal separable Baach space, a real valued fuctio f o X, which is Gâteaux-differetiable at every poit, has bouded o-empty support, ad with the properties that f is orm to wea cotiuous ad f (X) has a isolated poit a, ad that ecessarily a Mathematics Subject Classificatio : primary : 46 B 20, secodary :. Key words : Gâteaux-derivatives, bump fuctios, geometry of Baach spaces. 1 Supported by grats A ad GAČR 201/01/

2 2 1) Itroductio. Let f be a mappig from a Baach space X ito a Baach space Y which is Gâteaux-differetiable at every poit. Our purpose is the study of the rage of the derivative of f. We deote this rage f (X). Let us recall that sufficiet coditios o a subset A of a dual Baach space X so that it is the rage of a real valued fuctio o X which is Frechet-differetiable at each poit have bee obtaied i [BFKL], [BFL], [AFJ] ad [G1]. I this case, it was oticed i [AD] that wheever X is a ifiite dimesioal Baach space with separable dual, there exists a C 1 -smooth real valued fuctio o X with bouded support ad such that f (X) = X. O the other had, it follows from [H] that if f is a fuctio o c 0 with locally uiformly cotiuous derivative, the f (c 0 ) is icluded i a coutable uio of orm compact subsets of l 1. The structure of the rage of f wheever f satisfies a Holder coditio has bee ivestigated i [G2]. I the case of fuctios or mappigs which are Gâteaux-differetiable at each poit, it was observed i [ADJ] that f (X) ca coicide with L(X,Y ). We shall ivestigate here pheomea which ca occur whe f is Gâteauxdifferetiable, but ot whe f is Frechet-differetiable. I particular, for each ifiite dimesioal separable Baaach space X, we shall costruct i sectio 2 a Gâteaux-differetiable fuctio f o X, with bouded support, ad such that for all x 0, f (x) f (0) 1. I sectio 3, we shall cosider the followig questio : let X,Y be two Baach spaces. Is it possible to costruct a Lipschitz cotiuous mappig f : X Y, Gâteauxdifferetiable at each poit, ad such that, for all x,y X, x y, we have f (x) f (y) 1? Clearly, this is ot possible wheever L(X,Y ) is separable. We shall prove that this is ot possible either wheever Y = IR, but such a costructio will be carried out wheever (X,Y ) = (l 1,IR 2 ) ad wheever (X,Y ) = (l p,l q ) with 1 p q < +. 2) Isolated poits i the rage of the derivative of a fuctio. Let X be a Baach space, ad f be a real valued fuctio defied o X. If f is Frechet-differetiable at every poit, the Maly s Theorem asserts that the rage of f, deoted f (X), is coected. If f is Gâteaux-differetiable at every poit of X ad if f is orm to wea cotiuous, the f (X) is wea coected. Therefore, if f is ot affie, o poit of f (X) is isolated i f (X) edowed with the wea -topology. This result remais true eve if f is ot assumed to be orm to wea cotiuous, as show by the followig propositio. We shall see later that i this case f (X) is ot ecessarily orm coected. Propositio : Let X be a ifiite dimesioal Baach space, ad let f be a real valued locally Lipschitz ad Gâteaux-differetiable fuctio o X. The either f is affie, or, for every x X, f (x) lies i the wea closure of f (X)\{f (x)}.

3 3 Remar : J. Sait Raymod costructed a mappig f from IR 2 ito IR 2, Frechet-differetiable at each poit, ad so that { det(f (x));x IR 2} = {0,1}. Therefore f (IR 2 ) is ot coected ad for every x IR 2, f (x) / f (X)\{f (x)}. Cosequetly, there is o aalog of Maly s theorem ad of the above propositio for vector valued mappigs. Proof : Let f be a real valued locally Lipschitz ad Gâteaux-differetiable fuctio o X which is ot affie. Therefore, Card ( f (X) ) 2. I order to get a cotradictio, assume moreover that f (X) = A {y}, where A ad y / A w. Sice y f (X), there exists x X such that y = f (x). Replacig f by f(x +.), we ca assume that x = 0. Fix also x 0 X such that f (x 0 ) A. Sice y / A w, there exists x 1,x 2,...,x X ad ε > 0 such that, if we deote ỹ = ( y(x 1 ),y(x 2 ),...,y(x ) ) IR ad à = {( z(x 1 ),z(x 2 ),...,z(x ) ) ;z A } IR the, for every z Ã, z ỹ > ε. If we deote ỹ = ( y(x 0 ),ỹ ) IR +1 ad à = ( z(x 0 ),z(x 1 ),z(x 2 ),...,z(x ) ) ;z A } IR +1, the we also have that, for every z Ã, z ỹ > ε. Defie F : IR +1 IR by F(t 0,t 1,t 2,...,t ) = f ( ) t i x i Sice F is Lipschitz cotiuous ad Gâteaux-differetiable o IR +1, it is Fréchet-differetiable o IR +1 ad F (t 0,t 1,t 2,...,t ) = ( f ( i=0 i=0 t i x i ) (xj ) ) j=0 à {ỹ} Moreover F (0,0,...,0) = ỹ, F (1,0,...,0) Ã. Therefore F (IR +1 ) is ot coected ad this cotradicts the Theorem of Maly. From ow o, we say that a real valued fuctio o a ifiite dimesioal Baach space X is a bump fuctio if it has bouded o empty support. We shall deote B(r) the set of all x X such that x < r. If E is a Baach space, x E ad r > 0, we deote B E (x,r) (resp. B E (x,r)) the ope ball (resp. closed ball) i E of ceter x ad radius r. If f is a cotiuous ad Gâteaux-differetiable bump fuctio o X, the, accordig to the Eelad variatioal priciple, the orm closure of f (X) cotais a ball B(r) for some r > 0. A atural cojecture would be that the orm closure of f (X) is orm coected, or at least that f (X) does

4 4 ot cotai a isolated poit. This is ot so as show by the followig costructio. Theorem 1 : Let X be a ifiite dimesioal separable Baach space. The, there exists a bump fuctio f o X such that f is Gâteaux-differetiable at every poit, f is orm to wea cotiuous ad f (0) f (x) 1 wheever x 0. If X is separable, we ca assume moreover that f is C 1 o X\{0}. Remar : Accordig to the above discussio, 0 is ot a isolated poit of f (X), so ecessarily f (0) 0. Proof : We shall use two lemmas. Lemma 1 : Let X be a Baach space, U be a ope coected subset of X such that 0 U ad x U. Assume there exists o X a Lipschitz cotiuous bump fuctio which is Gâteaux-differetiable (resp. Frechetdifferetiable) at every poit. The there exists a Lipschitz cotiuous bump fuctio β o X which is Gâteaux-differetiable (resp. Frechetdifferetiable) at every poit, such that β (X) U ad β (x) = x for all x i a eighbourhood of 0. Proof of lemma 1 : Sice U is coected, there exists fiitely may poits x 0,x 1,...,x U such that x 0 = 0, x = x, ad the segmets [x i,x i+1 ] are icluded i U. The polygoal lie R = 1 [x i,x i+1 ] is compact, therefore there exists ε > 0 such that R + B(ε) U. Let b be a Lipschitz bump fuctio o X which is Gâteaux-differetiable (resp. Frechetdifferetiable) at every poit of X. By traslatio, we ca assume that b(0) 0. Replacig b(x) by λ 1 b(λ 2 x), we ca also assume that there exists 0 < δ < 1 such that b(x) 1 wheever x δ ad that the support of b is icluded i the uit ball. Composig b with a suitable C -smooth fuctio from IR ito IR, we ca assume moreover that b(x) = 1 wheever x δ, ad that 0 b(x) 1 for all x X. By addig if ecessary poits o the polygoal lie R, we ca assume that for all i {1,2,...,}, x i x i 1 < ε/ b. Defie i=0 b i (x) = b(x).(x i x i 1)(x) We have b i (x) = (x i x i 1 )(x).b (x)+b(x).(x i x i 1 ), with b(x).(x i x i 1 ) [0,x i x i 1 ] ad (x i x i 1 )(x).b (x) < ε for all x X, therefore b i (X) [0,x i x i 1 ] + B(ε). Fially, set β(x) = δ i 1 ( b i x/δ i 1 ) i=1

5 β is a Lipschitz cotiuous bump fuctio o X which is Gâteaux-differetiable (resp. Frechet-differetiable) at every poit. Let x X ad assume that δ i < x δ i 1 for 1 i. If j > i, x/δ j 1 > 1, so b j ( y/δ j 1 ) = 0 for all y i a eighbourhood of x ad b j( x/δ j 1 ) = 0. If j < i, x/δ j 1 δ, so b j( x/δ j 1 ) = x j x j 1. Therefore i 1 β (x) = (x j x j 1) + b i( x/δ i ) = x i 1 + b i( x/δ i ) [x i 1,x i ] + B(ε) j=1 Moreover, if x δ, the β (x) = x = x. Thus β (x) = x for all x i a eighbourhood of 0 ad β (X) R + B(ε) U. Lemma 2 : Let X,Y be two Baach spaces, a X, V be a ope eighbourhood of a, ad f : V Y be cotiuous o V ad Gâteauxdifferetiable at every poit of V \{a}. If f (x) has a wea limit l as x teds to a, the f is Gâteaux-differetiable at a ad f (a) = l. Proof of lemma 2 : Fix h X. The mappig φ h defied o the real lie by φ h (t) = f(a + th) wheever t 0, φ h (t) = f (a + th).h teds to l.h as t teds to 0. Therefore f is differetiable at a i the directio h ad f (a).h = l.h. This proves that f is Gâteaux-differetiable at a ad f (a) = l. I order to prove the theorem, let a X such that 1 < a < 2. Let (u ) be a dese sequece i X ad V = { x X ; x (u i ) a(u i ) < 1/2 for all i {1,...,} } (V ) 0 be a decreasig sequece of wea ope subsets cotaig a so that, if y V ad if (y ) is bouded, the (y ) coverges to a for the wea -topology. Moreover, W = V {x X ;1 < x a < 2} is coected for each. Let (x ) X be a sequece such that x 1 = 0 ad for every, x W. For each, 1 < x a < 2 ad (x ) coverges to a for the wea topology. W x = {x x ;x W } is a orm ope coected subset of X cotaiig 0. Sice x +1 W +1 W, we also have x +1 x W x. Sice X is separable (resp. X is separable) there exists o X a Lipschitz cotiuous bump fuctio which is Gâteaux-differetiable (resp. Frechet-differetiable) at each poit. Accordig to lemma 1, there exists a Lipschitz cotiuous bump b which is Gâteaux-differetiable (resp. Frechet-differetiable) at every poit, such that b (X) W x, with support i the uit ball ad such that b (x) = x +1 x for all x satisfyig x < δ. Deote c 1 = 1 ad, for 2, c = 1 δ. Defie i=1 b(x) = + =1 c b ( x/c ) 5

6 6 b has bouded support sice b(x) = 0 wheever x 1. O X\{0} this sum is locally fiite, so b is Gâteaux-differetiable (resp. Frechetdifferetiable) at each poit of X\{0}. If δ x < δ +1, the we have b (x) = x + b (x) W, so b (x) is uiformly bouded i x, b (X\{0}) X \B(a,1), ad b (x) w a as x 0. Lemma 2 the shows that b is Gâteaux-differetiable at 0 ad that b (0) = a. 3) Ca all the derivatives be far away from each other? We first otice that, uder mild regularity assumptios, the aswer to the above questio is egative for fuctios. Propositio : Let X be a Baach space ad f : X IR be a Lipschitz cotiuous, everywhere Gâteaux-differetiable fuctio. The, for every x X ad every ε > 0, there exists y,z B X (x,ε) such that f (y) f (z) ε. Proof : We shall actually show that if f : X IR is locally uiformly cotiuous ad everywhere Gâteaux-differetiable, the, for every x X ad for every ε > 0, there exists δ > 0 such that for every h X, h δ, there exists y B X (x,ε) such that f (y + h) f (y) ε. Fix x X ad ε 0 > 0 such that f is uiformly cotiuous o B X (x,2ε 0 ). Fix also 0 < ε < ε 0. By uiform cotiuity, there exists δ > 0 such that f(z) f(y) < ε 2 /4 wheever y,z B X (x,2ε 0 ) ad z y δ. Without loss of geerality, we ca assume that δ < ε/2. Tae ay h X such that h δ. Defie ϕ : X IR by ϕ(y) = f(y + h) f(y) if y x ε 0 ad ϕ(y) = + otherwise. The fuctio ϕ is lower semicotiuous o X ad, for all y B X (x,ε 0 ), ε 2 /4 < ϕ(y) < ε 2 /4. I particular, ϕ(x) < if y X ϕ(y) + ε 2 /2. The Eelad variatioal priciple the tells us the existece of y X such that y x ε/2 ad for all u X, ϕ(u) ϕ(y) ε u y. Sice y x ε/2 < ε 0, the fuctio ϕ is Gâteaux differetiable at y ad we obtai ϕ (y) ε. Hece, if we deote z = y + h, f (y) f (z) ε, ad we have z x h + y x < ε. The derivatives of a Frechet differetiable mappig caot be far away from each other for mappigs which are everywhere Frechet-differetiable. Propositio : Let X,Y be separable Baach spaces ad f : X Y be a everywhere Fréchet-differetiable locally uiformly cotiuous mappig. The, for every x X ad every ε > 0, there exists y,z B X (x,ε), y z, such that f (y) f (z) ε. Proof : Fix ε > 0 ad 0 > 0 such that f is uiformly cotiuous o B X (x,ε + 1/ 0 ). For each 1, defie A = { y B X (x,ε), f(y+h) f(y) f (y).h ε h wheever h 1/ }

7 Sice B X (x,ε) = 0 A, there exists 1 0 ad u B X (x,ε) such that u is a accumulatio poit of A 1. Pic y,z A 1 such that y z ad y z < α, where α is chose so that f(u) f(v) ε/ 1 wheever u,v B(x,ε + 1/ 0 ) ad u v < α. We have 7 f(y+h) f(y) f (y).h ε/ 1 ad f(z+h) f(z) f (z).h ε/ 1 So, for all h such that h 1/ 1, ( f(y + h) f(z + h) ) ( f(y) f(z) ) ( f (y) f (z) ).h 2ε/1 Therefore, ( f (y) f (z) ).h 4ε/ 1 Sice this is satisfied for for all h such that h 1/ 1, we obtai that f (y) f (z) 4ε. I view of the above propositios, oe could believe that wheever X,Y are Baach spaces (or vector ormed spaces) ad f : X Y is a mappig Gâteaux-differetiable at each poit of X, the for every ε > 0, there exists y,z X such that f (y) f (z) ε. Our ext result proves that this is ot so. Theorem 2 : 1) There exists a Lipschitz mappig F : l 1 IR 2, Gâteauxdifferetiable at each poit of l 1, such that for every x,y l 1, x y, the F (x) F (y) L(l 1,IR 2 ) 1. Moreover, for each h l 1, x F (x).h is cotiuous from l 1 ito IR 2. 2) Let us deote D the vector ormed space of elemets of l 1 with fiite support. There exists a Lipschitz fuctio G : l 1 IR, Gâteauxdifferetiable at each poit of l 1, such that for every x,y D, x y, the G (x) G (y) l 1. We shall costruct F ad G with the properties of theorem 2 usig series. We were ispired by a costructio from [DI]. We eed a auxiliary costructio. Lemma 3 : Give = (a,a,b,b ) IR 4 such that a < a < b < b ad ε > 0, there exists a C -fuctio ϕ = ϕ,ε : IR 2 IR 2 such that : (i) ϕ(x,y) ε for all (x,y) IR 2, (ii) ϕ(x,y) = 0 wheever x / [a,b ], (iii) ϕ x (x,y) ε for all (x,y) IR 2, (iv) ϕ y (x,y) = 1 wheever x [a,b], (v) ϕ y (x,y) 1 for all (x,y) IR 2,

8 8 (vi) If we deote ϕ(x,y) = ( ϕ 1 (x,y),ϕ 2 (x,y) ), the ϕ 1 y (x,0) = 1 wheever x [a,b]. Proof of Lemma 3 : Let b : IR IR be a C -smooth fuctio such that 0 b(x) 1 for all x, b(x) = 0 wheever x / [a,b ] ad b(x) = 1 wheever x [a,b]. If 1 is large eough, the fuctio defied by ϕ(x,y) = b(x) ( ) si(y),cos(y) satisfies the desired properties. We shall also use the followig criterium of Gâteaux-differetiability of the sum of a series : Lemma 4 : Let X ad Y be Baach spaces ad, for all, let f : X Y be Gâteaux-differetiable mappigs. Assume that ( f ) coverges poitwise o X, ad that there exists a costat K > 0 so that for all h, (1) sup x X 1 f h (x) K h The the mappig f = 1 f is Gâteaux-differetiable o X, for all x, f (x) = 1f (x) (where the covergece of the series is i L(X,Y ) for the strog operator topology), ad f is K-Lipschitz. Moreover, if each f is cotiuous from X edowed with the orm topology ito L(X,Y ) with the strog operator topology, the f shares the same cotiuity property. Proof of Lemma 4 : Fix x X. First observe that coditio (1) implies that for all h, the series ( f h (x)) = ( f (x).h ) coverges i Y. Therefore, the series ( f (x) ) coverges i L(X,Y ) for the strog operator topology, to some operator T L(X,Y ), ad by (1), T K. For each h X, we defie g : IR Y by g (t) = f (x + th). The fuctio g = g is well defied. Sice 1 1 g 1 sup x X f h (x) K h f h (x) = T(h). the mappig g is differetiable ad g (0) = g (0) = 1 1 Thus we have proved that f is differetiable alog every directio h ad that f h (x) = T(h). I other words, f is Gâteaux-differetiable at x ad f (x) = T. Sice for all x, f (x) K, the mea value theorem implies that f is K-Lipschitz. Proof of Theorem 2, part 1) : Fix a eumeratio = (a,a,b,b ), N, of all quadruples of dyadic umbers such that a < a < b < b.

9 Select itegers m such that for each, < m ad (m ) is a icreasig sequece, ad satisfyig (2) m = m p l = p ad = l Fix ε > 0 ad let ε be positive real umbers such that ε =1 =1 9 = ε. We shall otice ε = ε, so that ε = ε. Put f, : l 1 IR 2 such that, =1 ( ) =1 ) if x = (x i ) l 1, the f, x = ϕ,ε( x,x m : f, is a C fuctio o l 1. The fuctio F : l 1 IR 2 we are looig for is defied by : F(x) = f, (x) IN IN Claim 1 : F is well-defied. Ideed, accordig to coditio (i) of the lemma, f, = ϕ,ε = ε, so the series defiig F coverges uiformly. Claim 2 : F is Gâteaux-differetiable o l 1 ad F is (1 + ε)-lipschitzcotiuous o l 1. To see this, we apply Lemma 4 : let h=(h 1,...,h,...) l 1. By (iii) ad (v), we have for all, : sup f, x X h (x) h m + ε h h m + ε h 1 So, because of coditio (2),, sup x X f, h (x) (1 + ε) h 1 We have proved that coditio (1) of Lemma 4 is satisfied with K = 1+ε, thus F is Gâteaux-differetiable o l 1 ad F is (1+ε)-Lipschitz-cotiuous o l 1. Claim 3 : If x y l 1, the F (x) F (y) L(l 1,IR 2 ) 1 2ε. Ideed, let IN such that x y. Let such that x [a,b ] ad y / [a,b ]. Accordig to (ii) ad (iv) of Lemma 3, f, f, (x) = 1 ad (y) = 0 x m O the other had, for all r, ad, if l m f m,r (x) ε r ad (l,r) (,), ad f m,r (y) ε r f l,r (x) = 0 ad f l,r (y) = 0

10 10 Therefore, F (x) F (y) L(l1,IR 2 ) F (x) 1 1 2ε (l,r) =(,) F f l,r (y) (x) f l,r (y) Let us ow prove part 2) of Theorem 2. Sice F : l 1 IR 2, we ca write F = (G,H), where G,H : l 1 IR. We shall also deote f, = (g,,h, ). G : l 1 IR is Lipschitz cotiuous, Gâteaux-differetiable at each poit of l 1. Let x = (x i ),y = (y i ) D ad such that x y. Let such that x [a,b ], y / [a,b ] ad x m = 0. Accordig to (vi) of Lemma 3, we have g, (x) = 1 ad g, (y) = 0 We coclude, as i the proof of Claim 3 of part 1), that G (x) G (y) l 1 2ε Remar : 1) If we set Φ = f, we have obtaied for every α > 0, the 1 2ε costructio of a fuctio Φ : l 1 IR 2, Gâteaux-differetiable at every poit of l 1, satisfyig : (i) for all x,y l 1, Φ(x) Φ(y) (1 + α) x y 1, (ii) for all x y l 1, Φ (x) Φ (y) L(l 1,IR 2 ) 1. 2) Fix h l 1. Sice x F (x).h is cotiuous from l 1 ito IR 2, the set { F (x).h; x l 1} is coected. This is i cotrast with the fact that { F (x) ; x l 1} is discrete i L(l 1,IR 2 ). 3) A carefull loo at the above costructio shows that f is uiformly Gâteaux-differetiable. 4) Observe that for cardiality reasos, wheever L(X, Y ) is separable, the for every Gâteaux-differetiable mappig from X ito Y, ad for every ε > 0, there exists y,z X such that f (y) f (z) ε. Therefore, it is ot possible to replace l 1 by l p (p > 1) i Theorem 2. However, there exists a Lipschitz fuctio H : l 2 l 2, Gâteaux-differetiable at each poit of l 2, such that for every x,y l 2, if x y, the H (x) H (y) L(l 2 ) 1 This will follow from the followig more geeral result :

11 Theorem 3 : Let X p = l p if 1 p < + ad X = c 0. Let us fix 1 p, q +. The followig assertios are equivalet : (1) There exists a Lipschitz fuctio H : X p X q, Gâteaux-differetiable at each poit of X p, such that for every x,y X p, x y, the H (x) H (y) L(Xp,X q ) 1. (2) p q. (3) L(X p,x q ) is ot separable. Proof of Theorem 3 : Accordig to Remar 4) above, (1) implies (3). If p > q, the by Pitt s theorem, all operators from X p to X q are compact, hece L(X p,x q ) is separable. Therefore (3) implies (2). So it remais to prove that (2) implies (1). Assume that p q ad let (e ) be the usual basis of X p. Let T L(IR 2,X q ) defied by T (x,y) = xe 2 + ye 2+1. Deote a q the commo orm of the operators T. Let, ε, m ad ϕ,ε defied as i the proof of Theorem 2. Put f, : X p X q such that, ( ) if x = (x i ) X p, the f, x = Tm ϕ ( ),ε x,x m : the fuctios f, is a C mappig from X p ito X q. The fuctio H : X p X q we are looig for is defied by : H(x) = f, (x) IN IN As i the proof of Theorem 2, H is well-defied. Lemma 4 is o loger applicable i order to show that H is Gâteaux-differetiable at each poit of X p. But lemma 4 remais true if the hypothesis (1) from lemma 4 is replaced by coditio (2) below : (2) for all h, ( f h (x)) coverges uiformly with respect to x So, fix h=(h 1,...,h,...) X p. We have f, h (x) = h u, (x) + h m v,(x) with u, (x) q ε a q, v, (x) spa{e 2m,e 2m +1 } ad v, (x) q a q. We claim that both series ( h u, (x) ) ad ( h m v,(x) ) are uiformly covergig with respect to x. Ideed, for the first oe, this fol-, lows from the fact that for each x, h u,m (x) q h p.a q.ε, ad that ε < +. For the secod oe, ( h m v,m(x) ) coverges uiformly because it satisfies the uiform Cauchy coditio. Ideed, fix δ > 0 =1 =1 ad a fiite set A IN IN such that h p m < δ p. For fixed x, (,)/ A the v, (x) are elemets of X q with disjoit supports, so, for ay fiite subset F of (IN IN)\A, (,) F h m v,m(x) Xq = ( (,) F a q ( (,) F h m v,m(x) q X q ) 1/q h q m ) 1/q aq ( (,) F 11 h p m ) 1/p < aq δ

12 12 Notice that we used i the above chai of iequalities the fact that p q. The above estimate is uiform i x, therefore the series ( h m v,m(x) ) satisfies the uiform Cauchy coditio. Applyig the variat of lemma 4 metioed above, we get that H is Lipschitz cotiuous ad Gâteauxdifferetiable at each poit of l 2. As i the proof of theorem 2, oe sees that there exists a > 0 such that for every x,y l 2, if x y, the H (x) H (y) L(l p,l q ) a. Refereces [AD] D. Azagra ad R. Deville, James theorem fails for starlie bodies, J. Fuct. Aal. 180 (2001), [ADJ] D. Azagra, R. Deville ad M. Jimeez-Sevilla, O the rage of the derivatives of a smooth fuctio betwee Baach spaces, to appear i Math Proc.. [AFJ] D. Azagra, M. Fabia ad M. Jimeez-Sevilla, Exact fillig of figures with the derivatives of smooth mappigs betwee Baach spaces, to appear. [BFKL] J. M. Borwei, M. Fabia, I. Kortezov ad P. D. Loewe, The rage of the gradiet of a cotiuously differetiable bump, J. Noliear ad Covex Aal. 2 (2001), [BFL] J. M. Borwei, M. Fabia ad P. D. Loewe, The rage of the gradiet of a Lipschitzia C 1 -smooth bump i ifiite dimesios, to appear. [DI] R. Deville ad M. Ivaov, Smooth variatioal priciple with costraits, Math. Nachrite 69 (1997) [G1] T. Gaspari, O the rage of the derivative of a real valued fuctio with bouded support, to appear. [G2] T. Gaspari, Bump fuctios with Holder derivatives, to appear. [H] P. Háje, Smooth fuctios o c 0, Israel J. Math., 104 (1998), [M] J. Maly, The Darboux property for gradiets, Real Aal. Exch. 22, No.1, (1996). [M] J. Sait-Raymod, Local Iversio for Differetiable Fuctios ad Darboux Property, to appear i Mathematia.