Rolle s Theorem and Negligibility of Points in Infinite Dimensional Banach Spaces*

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 3, 8795 997 ARTICLE NO. AY97555 Rolle s Theorem and Negligibility of Points in Infinite Dimensional Banach Spaces* D. Azagra, J. Gomez, and J. A. Jaramillo Departamento de Analisis Matematico, Facultad de Ciencias Matematicas, Uniersidad Complutense, Madrid, 800, Spain Submitted by Richard M. Aron Received July 7, 996 In this note we prove that if a differentiable function oscillates between and on the boundary of the unit ball then there exists a point in the interior of the ball in which the differential of the function has norm equal or less than. This kind of approximate Rolle s theorem is interesting because an exact Rolle s theorem does not hold in many infinite dimensional Banach spaces. A characterization of those spaces in which Rolle s theorem does not hold is given within a large class of Banach spaces. This question is closely related to the existence of C diffeomorphisms between a Banach space X and X 0 which are the identity out of a ball, and we prove that such diffeomorphisms exist for every C smooth Banach space which can be linearly injected into a Banach space whose dual norm is locally uniformly rotund Ž LUR.. 997 Academic Press. INTRODUCTION Rolle s theorem in finite dimensional spaces states that for every open connected and bounded subset U in n and every continuous function f : U such that f is differentiable in U and constant on U, there exists an x in U such that dfž x. 0. In a paper published in 99, S. A. Shkarin 0 proved that Rolle s theorem fails in a large class of infinite dimensional Banach spaces, including all super-reflexive and all non-reflexive Banach spaces having a Frechet differentiable normalthough he did not study the reflexive but non-super-reflexive case. Other explicit examples *Supported in part by DGICYT PB9305. E-mail address: daniel@sunam.mat.ucm.es. E-mail address: javier@sunam.mat.ucm.es. E-mail address: jaramil@eucmax.sim.ucm.es. 87 00-7X97 $5.00 Copyright 997 by Academic Press All rights of reproduction in any form reserved.

88 AZAGRA, GOMEZ, AND JARAMILLO were found in c and l by J. Ferrera and J. Bes 0 7 and independently by J. Ferrer 8. On the other hand it is clear that Rolle s theorem trivially holds in all non-asplund Banach spaces because of the harmonic behaviour of differentiable maps in such spaces. It is natural to conjecture that a reasonable version of Rolle s theorem in infinite dimensional Banach spaces holds if and only if our space does not have a C bump function and we prove this conjecture to be true within the class of those Banach spaces X which can be linearly injected into a Banach space Y with an equivalent norm whose dual norm is locally uniformly rotund Ž LUR. in Y *. This geometrical condition, which we shall call Ž. for short, is satisfied by every Ž WCG. Banach space, every space which can be injected into some c Ž. 0, and even by every space injectable into some Ž. C K, where K is a scattered compact with K. This conjecture is closely related to the question posed in whether for every Banach space X having a C bump function there exists a C diffeomorphism : XX0 such that is the identity out of a ball. We give an affirmative answer to this question within the class of all Banach spaces X verifying Ž.. An interesting approximate version of Rolle s theorem remains nevertheless true in all Banach spaces, as we prove in this note. By an approximate Rolle s theorem we mean that if a differentiable function oscillates between and on the boundary of the unit ball then there exists a point in the interior of the ball in which the differential of the function has norm less than or equal to. The authors gratefully acknowledge their debt to Juan Ferrera, who called their attention to this kind of problem. They are also grateful to Jesus Ferrer for pointing out some inaccuracies in the original version of this note.. THE APPROXIMATE ROLLE S THEOREM In order to prove the approximate Rolle s theorem we need the following lemmas, which are themselves interesting. LEMMA.. Let X be a Banach space and U be an open bounded connected subset of X. Let f : U be a continuous bounded function such that: Ž. f is Gateaux ˆ differentiable in U Ž. inf fž U. inf fž U. or sup fž U. sup fž U.. Then, for eery 0 there exists x U such that dfž x..

ROLLE S THEOREM 89 Proof. We may suppose inf fž U. inf fž U.. Let us choose x0 U such that fž x. inf fž U., and let, be such that 0 inf fž U. 0 fž x. and 0 R, where R supx x : x U 0 0. From Ekeland s Variational Principle Žsee Lemma 3.3 in 9, or. 5 it follows that there exists x U such that for all x x. In particular fž x. fž x. xx Ž. fž x. fž x. x x fž x. Rinf f Ž U. 0 0 0 and therefore x U. On the other hand, inequality every h such that h, Ž. implies that for fž xth. fž x. df Ž x.ž h. lim, t0 t which proves dfž x.. LEMMA.. Let X be a Banach space and U be an open bounded connected subset of X. Let f : U be a continuous bounded function such that: Ž. f is Gateaux ˆ differentiable in U Ž. fž U. a, b, where a b. Then, for eery x U and R 0 such that BŽ x, R. 0 0 U, there exists x BŽ x, R. such that dfž x. Ž b a. 0 R. Proof. We may suppose that a, b,. Two cases will be considered. Case I. fž x. 0. We may suppose fž x. 0 Žthe case fž x. 0 0 0 0is analogous.. From Ekeland s Variational Principle Žsee Lemma 3.3 in 9, or. 5 it follows that there exists x U such that Ž. x x ŽfŽ x.. Ž R. 0 0 R, and Ž. fž x. fž x. Ž R.xx for all x x. Ž. Ž. From we get x U and implies that for every h with h fž xth. fž x. df Ž x.ž h. lim R, t0 t which proves df x R.

90 AZAGRA, GOMEZ, AND JARAMILLO Case II. fž x. 0. We may suppose dfž x. 0 0 R, since we would have finished otherwise. If dfž x. R there exists h with h 0 such that dfž x.ž h. 0 R and therefore there exists 0 such that fž x h. 0 R. Applying Ekeland s Variational Principle again we obtain x U such that: Ž. x Ž x h. ŽfŽ x h.. Ž R. Ž 0 0 R. Ž R. R and Ž. fž x. fž x. Ž R.xx for all x x. From Ž. it follows that x x x Ž x h. 0 0 R, so that x BŽ x, R. U, and Ž. implies dfž x. R. 0 The following result is immediately deduced as a consequence of Lemmas. and.. THEOREM.3 Ž Approximate Rolle s Theorem.. Let X be a Banach space and U be an open bounded connected subset of X. Let f : U be a continuous bounded function. Suppose that f is Gateaux ˆ differentiable in U and f Ž U. a, b, with a b. Then, for eery R 0 and x0 U such that BŽ x, R. U, there exists x U such that 0 ba df Ž x.. R From this we can immediately deduce the following COROLLARY.. Let U be an open connected bounded subset of a Banach space X. Let f : U: be continuous, bounded, and Gateaux ˆ differentiable in U. Suppose that f is constant on U. Then, inf fž x. 0. xu It is easy to see, using Ekeland s Variational Principle, that if X is a Banach space and f : X is continuous, Gateaux ˆ differentiable, and bounded below Ž or bounded above., then inf fž x. x X 0. Alterna- tively, if we assume that f is bounded, this is an immediate consequence of Theorem.3. 3. DIFFEOMORPHISMS BETWEEN X AND X 0 We use in this section Bessaga s non-complete norm technique to prove that every Banach space X verifying the condition Ž. There exists a Banach space Y with an equivalent norm. whose dual norm. * is locally uniformly rotund Ž LUR. in Y * and a continuous linear injection T : X Y

ROLLE S THEOREM 9 is C diffeomorphic to X 0. If moreover X has a differentiable bump function then there exits a C diffeomorphism : X X 0 such that is the identity out of a ball centered at 0. It is not difficult to see that condition Ž. is equivalent to saying that X admits a continuous Ž not necessarily equivalent. norm whose dual norm is Ž LUR.. Recall that a norm in a Banach space ŽX,.. is said to be non-complete provided the normed space Ž X,. is not complete. THEOREM 3.. Let X be an infinite dimensional Banach space that erifies condition Ž.. Then Ž. X admits a C ŽX 0. non-complete norm ; Ž. there exists a C diffeomorphism : X X 0 such that Ž x. xifž x.. Proof. First of all let us see that every Banach space Y with an equivalent norm. whose dual norm. * is Ž LUR. admits a C ŽY0. non-complete norm. It is known that every infinite dimensional Banach space admits a continuous non-complete norm Žsee, Chap. III, Lemma 5... Let g : Y be such a norm in Y. Define Ž y. inf g Ž u. yu : uy, yy. It is easy to check that is a continuous norm in Y. As Ž y. gž y. for all y Y and g is non-complete, it is obvious that is also noncomplete. On the other hand, it is known Žsee 6, Proposition.3. that if ŽY,.. is a Banach space such that the dual norm. * is LUR then for every proper convex lsc function f : Y Ž, the infimal convolution with the squared norm f Ž y. inf fž u. nyu : uy, yy, n is C smooth and convex and if moreover f is bounded on bounded sets,. then fn f uniformly on bounded sets as n. Taking f g and n, from this result we obtain that is C Ž Y., so that is C ŽY0.. Now we should note that every subspace Z in Y has an equivalent norm whose dual norm is LUR. Indeed, considering the projection : Y * Z*, Ž y*. y, and using Theorem.Ž ii. of 3, Chap. II Z, we get that Z* has an equivalent LUR dual norm. Therefore, if Y * has an equivalent LUR dual norm, then every closed subspace of Y admits a C non-complete norm. Ž

9 AZAGRA, GOMEZ, AND JARAMILLO So let X, Y, and T : X Y be as in condition Ž. and consider ZTŽ X..IfT X Z, since Z has a C non-complete norm and Ž. T : XZ is a linear isomorphism, 0 x T x defines a C noncomplete norm on X. If, on the contrary, TŽ X. is a dense but not closed subspace of Z, it is clear that 0 x T x defines a C non-complete norm on X. In any case we get a C non-complete norm on X. This proves Ž.. Now one can prove Ž. using Bessaga s non-complete norm technique as T. Dobrowolski does in. In fact Ž. is immediately deduced from Theorem 3.3 in. Nevertheless we will say a few words about the way one can construct the diffeomorphism. There exists a linearly independent sequence y in X such that Ý k6 Ž y y. k k k k k, where y 0, and a point y in the completion of Ž X,. such that y X and lim k yk y 0. Let : 0, be a C function with in Ž,, Ž. 0,., and. Define p : Ž 0,. X by k p t yý t ykyk k for t 0. p is a C path satisfying Žpt ps. ts, lim pt Ž. y,žpt Ž.. for all t 0 and pt Ž. t 0 0 if and only if t. Let x be an arbitrary vector in X and let F : 0,. 0,. be defined by FŽ. Žx pž.. for 0 and FŽ 0. Ž x y.. We have FŽ. FŽ., so from Banach s contraction principle applied to the interval 0,., it follows that the equation FŽ. has a unique solution. This means that for any x X, a number Ž x. with the property Ž. x p Ž x. Ž x. is uniquely determined. Moreover, since x, being in X, cannot be equal to y, we have x 0. This implies that the mapping Ž z. pž Ž z.. z is one-to-one from X 0 onto X, with Ž x. x pž Ž x... As and p are C,sois. Let Ž x,. Žx pž... Since for any x X we have x pžž x.. 0, the mapping is differentiable on a

ROLLE S THEOREM 93 neighbourhood of any point in X 0,. On the other hand, Ž x,. 0 because FŽ. FŽ.. So, using the implicit function theorem we obtain that : X 0 X is a C diffeomorphism. Finally, it is clear that Ž z. z whenever Ž z.. THEOREM 3.. For a Banach space X satisfying condition Ž., the following are equialent. Ž. X has a C bump function. Ž. There exists a C diffeomorphism : X X 0 such that is the identity out of a ball centered at 0. Proof. If : X X 0 is a C diffeomorphism such that Ž x. x whenever x r for some r 0, then, taking p X* such that pžž 0.. Ž. 0 and defining f x p x x we obtain a C bump function f such that fž. 0 0 and fž x. 0if xr, which proves that Ž. implies Ž.. Now suppose that X has a C bump function. Proposition 5. in 3, Chap. II gives us a function on X such that is C smooth on X 0, Ž tx. t Ž x. for x X and t, and there are constants a 0 and b0 such that axž x. bx for x X. Let : Ž 0,. Ž 0,. be a non-decreasing C function such that Ž. t 0 for t and Ž. t for t. Let Ž x. HŽ x. Ž Ž x.. Ž Ž x.. x, Ž x. for x 0, and HŽ. 0 0. H is a one-to-one mapping from X onto X transforming the set x X: Ž x. onto x X: Ž x., and H is C. Using the implicit function theorem as in the preceding theorem we obtain that H is also C. By composing this diffeomorphism with that of Theorem 3. we get a C diffeomorphism between X and X 0 that is the identity out of a ball centered at 0. Now we can prove as in the following COROLLARY 3.3. If a Banach space X erifies condition Ž. and has a Frechet smooth equialent norm then the sphere SX is C diffeomorphic to each hyperplane in X. If moreoer X is isomorphic to one of its hyperplanes, then X is C diffeomorphic to its sphere.

9 AZAGRA, GOMEZ, AND JARAMILLO Remark 3.. All the results in this section remain true if we replace condition Ž. by the following one Ž. There exist a Banach space Y with an equivalent differentiable norm., an infinite-dimensional reflexive closed subspace Z Y, and a continuous linear injection T : X Y such that Z TŽ X. Y. Indeed, if X, Y, and Z are as in this condition, let us consider any continuous noncomplete norm on Z and let us define 0 Ž x. inf w Ž z. xz : zz, xx. 0 Since Z is reflexive the infimum defining Ž x. is attained and using the differentiability of. it is easy to see that the norm is differentiable in Y. Moreover is non-complete because Ž z. Ž z. 0 for all z Z and is non-complete on the closed subspace Z Y. Now define Ž x. 0 ŽTŽ x.. for each x X. It is clear that is a differentiable noncomplete norm in X, and so we can construct the diffeomorphisms between X and X 0 in the same way as before.. AN EXACT ROLLE S THEOREM IN INFINITE DIMENSIONAL BANACH SPACES FAILS In this section we use the preceding results to prove that an exact Rolle s theorem either fails or trivially holds in infinite dimensional Banach spaces verifying Ž.. The following result, whose proof is clearly motivated by Shkarin s ideas in 0, provides a characterization of spaces that do not verify Rolle s theorem within the class of those spaces verifying Ž.. THEOREM.. If a Banach space X erifies condition Ž., the following are equialent: Ž. X has a C bump function. Ž. There exists an open connected bounded subset U and a continuous bounded function f : U such that f is C Ž U., f 0 on U, and yet dfž x. 0 for all x U; that is, Rolle s theorem fails in X. Ž. 3 There exists a C Ž X. bounded function f : X and an open connected bounded subset U in X such that f 0 on X U and yet dfž x. 0 for all x U. Proof. It is obvious that Ž. 3 implies Ž. and one can easily check that Ž. implies Ž.. Let us prove that Ž. implies Ž. 3. By Proposition 5. in 3, Chap.

ROLLE S THEOREM 95 II, there exists a function on X such that is C smooth on X 0, Ž tx. t Ž x. for x X and t, and there are constants a 0 and b0 such that axž x. bx for x X. From Theorem 3. we get a C diffeomorphism : X X 0 such that is the identity out of a ball centered at 0. Let : be an even C function such that 0, Ž. Ž. t 0 for all t Ž 0,. and Ž. t 0 for all t. We define f : X by f. Since f is the composition of the C functions : XX 0, : X 0 and, f is C, and f is bounded because so is. Moreover, we have fž x. 0ifŽŽ x... However, fž x. 0 for all x such that ŽŽ x.., because fž x.ž y. Ž Ž x.. dž Ž x..ž Ž x.ž y.. 0 for some y X since Ž x. is a linear isomorphism, d Ž z. 0 for all z X 0 and ŽŽŽ x... 0 whenever ŽŽ x... So, taking U xx:žž x.., Ž. implies Ž. 3 is proved. Remark.. Rolle s theorem trivially holds in non-asplund Banach spaces: if X is a non-asplund Banach space, U is an open connected bounded subset in X, and we have a continuous bounded function f : U that is Frechet differentiable in U and f 0 on U, then necessarily f 0onUŽsee 3, Chap. III, p. 97.. REFERENCES. C. Bessaga, Every infinite-dimensional Hilbert space is diffeomorphic with its unit sphere, Bull. Acad. Polon. Sci. Ser. Sci. Math. Ž 966., 73.. C. Bessaga and A. Pelczynski, Selected topics in infinite-dimensional topology, in Monografie Matematyczne, PWN, Warsaw, 975. 3. R. Deville, G. Godefroy, and V. Zizler, Smoothness and renormings in Banach spaces, in Pitman Monographs and Surveys in Pure and Applied Mathematics, Vol. 6, Longman, Harlow, 993.. T. Dobrowolski, Smooth and R-analytic negligibility of subsets and extension of homeomorphism in Banach spaces, Studia Math. 65 Ž 979., 539. 5. I. Ekeland, Nonconvex minimization problems, Bull. Amer. Math. Soc. ( N.S. ), No. 3 Ž 979., 37. 6. M. Fabian, P. Hajek, and J. Vanderwerff, On smooth variational principles in Banach spaces, J. Math. Anal. Appl. 97 Ž 996., 537. 7. J. Bes and J. Ferrera, private communication. 8. J. Ferrer, Rolle s theorem fails in l, Amer. Math. Monthly 03, No. Ž 996., 665. 9. R. R. Phelps, Convex functions, monotone operators and differentiability, in Lecture Notes in Mathematics, Vol. 36, Springer-Verlag, BerlinNew York, 993. 0. S. A. Shkarin, On Rolle s theorem in infinite-dimensional Banach spaces, Mat. Zametki 5, No. 3 Ž 99., 836.