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Non-Linear Maps Between Subsets of Banach Spaces A NON-LINEAR MAPS BETWEEN SUBSETS OF BANACH SPACES A dissertation submitted to Kent State University in partial ful¯llment of the requirements for the degree of Doctor of Philosophy by Reema Sbeih December, 2009 Dissertation written by Reema Sbeih B.S., Birzeit University, West Bank, Palestine 2000 M.S., Youngstown State University, 2002 M.A., Kent State University, 2008 Ph.D., Kent State University, 2009 Approved by Dr. Per Enflo , Chair, Doctoral Dissertation Committee Dr. Andrew Tonge , Member, Doctoral Dissertation Committee Dr. Morley Davidson , Member, Doctoral Dissertation Committee Dr. Kenneth Batcher , Member, Outside Discipline Dr. Peter Tandy , Member, Graduate Representative Accepted by Dr. Andrew Tonge , Chair, Department of Mathematical Sciences Dr. John R. D. Stalvey , Dean, College of Arts and Sciences ii TABLE OF CONTENTS ACKNOWLEDGEMENTS . iv 1 INTRODUCTION . 1 2 BASIC DEFINITIONS AND NOTATION . 3 3 PROJECTIONS ONTO THE UNIT BALL IN LP SPACES. 7 4 SCALING DOWN MAPS BETWEEN S(l1) AND S(l1) . 14 5 SCALING DOWN MAPS BETWEEN S(l1) AND S(l2) . 24 6 SCALING DOWN MAPS BETWEEN S(l1) AND S(lq) as q ! 1 . 37 7 CONCLUSION AND FUTURE PROJECTS . 48 BIBLIOGRAPHY . 50 iii ACKNOWLEDGEMENTS I owe this work to the two people from whom I received my life's blood, my parents. My ultimate inspiration comes from my advisor, Dr. Per Enflo, the wonderful mathematician who guided through this incredible and long journey. iv CHAPTER 1 INTRODUCTION In this dissertation we study non-linear maps between subsets of Banach spaces. As a background for this study we should mention two areas of mathematical research: Geometry of Banach Spaces and Approximation Theory. One way to study and compare the geometry of di®erent Banach spaces is to study linear maps between the spaces and to study how much these maps distort distances between points. There is an extensive literature on this [2], [3], [4], [5]. Another less studied way to compare the geometry of di®erent Banach spaces is to study maps between di®erent subsets of the spaces. In this case the maps are usually non-linear and the subsets are often unit balls or unit spheres of Banach spaces [6], [7], [10], [11]. In Approximation Theory, the following problem is important: how well can ob- jects (points) from a set A be approximated by objects (points) from a set B? One important way to study this approximation is to ¯nd for every point in A the nearest point in B, i.e., to ¯nd the nearest point map, also called the metric projection, and investigate its properties [10], [12], [13]. The maps that we study in this dissertation, the scaling down projection and the scaling down maps, are, as we shall see, nearest point maps. The dissertation is organized as follows: In Chapter 2 we give the basic de¯nitions and notation that we use in the rest of the paper. We also show that the scaling down projection and the scaling down maps are nearest point maps. In Chapter 3, we show that the scaling down projection onto the unit sphere in any 1 2 Banach space has Lipschitz norm at most 2 and that 2 is not attained. We also show that in L1(0; 1), the scaling down projection has norm 2, and that the scaling down projection in L1(0; 1) is the best projection in the sense that all other projections have norm ¸ 2. We also give estimates for the norm of the scaling down projection in Lp for 1 < p < 2. In chapters 4, 5, and 6 we study scaling down maps between the unit sphere of 1 p ln , being in some sense the largest unit sphere, and the unit spheres of ln-spaces. In particular we study the Banach-Mazur norms of these maps. Our results show that the Banach-Mazur norms of these maps are much larger than the Banach-Mazur norms of the simplest linear maps between the spaces. CHAPTER 2 BASIC DEFINITIONS AND NOTATION In this chapter we give the basic de¯nitions and notation that we use in this paper. We also show that the scaling down projection and the scaling down map are nearest point maps. De¯nition. [17] A vector space X is said to be a normed space if for every x 2 X there is associated a nonnegative real number jjxjj, called the norm of x, in such a way that (a) jjx + yjj · jjxjj + jjyjj for all x and y in X, (b) jj®xjj = j®jjjxjj if x 2 X and ® is a scalar, (c) jjxjj > 0 if x 6= 0. Every normed space maybe regarded as a metric space, in which the distance d(x; y) between x and y is jjx ¡ yjj. De¯nition. [17] A Banach space is a normed space which is complete in the metric de¯ned by its norm; this means that every Cauchy sequence is required to converge. Let (X; d) be a normed space. De¯nition. The unit ball of X, denoted by B(X) is B(X) = fx 2 X : d(x; 0) · 1g : De¯nition. The unit sphere of X, denoted by S(X) is S(X) = fx 2 X : d(x; 0) = 1g : 3 4 De¯nition. A projection is a map P : X ¡! X such that P ± P = P: De¯nition. We say that M ⊆ X is contractive if there is a projection P (in general nonlinear) from X onto M such that d(P x; P y) · d(x; y) for all x; y 2 X: De¯nition. Let M; N ⊆ X. A map T : M ¡! N is a nearest point map if d(x; T x) · d(x; z) for all z 2 N: De¯nition. A map P on X is called a Lipschitz map if there is a constant C such that (2.1) jjP x ¡ P yjj · Cjjx ¡ yjj for all x; y 2 X: The smallest constant C that satis¯es (2.1) is called the Lipschitz norm of P , and jjP x ¡ P yjj is denoted by jjP jj. It is easy to see that jjP jj = sup : x;y2X jjx ¡ yjj De¯nition. The map P on X de¯ned by 8 x <> if jjxjj > 1 P x = jjxjj > : x if jjxjj · 1 is called the scaling down projection. Consider the norms jj ¢ jj1 and jj ¢ jj2 on a vector space. Assume jjxjj1 · jjxjj2 for every x. Let X, Y be the normed spaces associated with jjxjj1 and jjxjj2 respectively. We have the following de¯nition. 5 De¯nition. The scaling down map T : S(X) ! S(Y ) is the map T that takes x 7! γxx where jjxjj1 = 1 and jjγxxjj2 = 1. Lemma 2.1 The scaling down projection is a nearest point map from X onto B(X). Proof. Let x 2 X and let y = P x. If jjxjj · 1, then y = x and jjx ¡ yjj = 0. x So suppose that jjxjj > 1; then y = . jjxjj We need to show that if z is any point in B(X) then jjx ¡ yjj · jjx ¡ zjj. We have ¯¯ ¯¯ ¯¯ µ ¶¯¯ µ ¶ ¯¯ x ¯¯ ¯¯ 1 ¯¯ 1 jjx ¡ yjj = ¯¯x ¡ ¯¯ = ¯¯x 1 ¡ ¯¯ = 1 ¡ jjxjj = jjxjj ¡ 1, and ¯¯ jjxjj¯¯ ¯¯ jjxjj ¯¯ jjxjj jjx ¡ zjj ¸ jjxjj ¡ jjzjj ¸ jjxjj ¡ 1. Therefore jjx ¡ yjj · jjx ¡ zjj for all z 2 B(X). ¥ Corollary 2.2 The scaling down map and its inverse are nearest point maps. De¯nition. A normed space X is strictly convex if jjtx1 + (1 ¡ t)x2jj < 1 whenever x1 and x2 are di®erent points of S(X) and 0 < t < 1. De¯nition. A normed space X is reflexive if every bounded sequence has a weakly convergent subsequence. De¯nition. A normed space X is smooth if every point on the unit sphere of X has a unique supporting hyperplane. Next we will give some of the notation used in this paper. R p p L (E) = ff : E jfj < 1g; 0 < p < 1: ¡R ¢ 1 p p jjfjjp = E jfj ; 0 < p < 1: jjfjj1 = inff® : m(fx 2 E : jf(x)j > ®g) = 0g: 1 L (E) = ff : jjfjj1 < 1g: 6 Let a = fakg be a sequence of real or complex numbers. Then we have the following de¯nitions: P 1 p p jjajjp = ( k jakj ) ; 0 < p < 1: jjajj1 = supkjakj: p l = fa : jjajjp < 1g; 0 < p < 1: 1 l = fa : jjajj1 < 1g: CHAPTER 3 PROJECTIONS ONTO THE UNIT BALL IN LP SPACES. It is known that unit balls in Hilbert spaces and spaces of continuous functions are contractive. In this chapter, we show that in any Banach space, the scaling down projection has Lipschitz norm · 2, and has norm equal to 2 in L1(0; 1). We also show that in L1(0; 1), any projection other than the scaling down projection has Lipschitz norm at least 2. We also estimate the norm of the scaling down projection in Lp(0; 1), 1 < p < 2. In a Hilbert space, a closed, convex set is contractive since the \nearest point map" is a contractive projection. Theorem 3.1 [1] (Beauzamy-Maurey result) Let X be reflexive, strictly convex, smooth Banach space of dimension 3 or greater. If the unit ball of X is contractive then X is a Hilbert space. Remark. Since l1 and C(0; 1) are not strictly convex, we cannot apply the Beauzamy- Maurey result. However, in C(0; 1) the unit ball is contractive. To see this, let f; g 2 C(0; 1). 7 8 Let 8 > > f if ¡1 · f · 1 <> P f = 1 if f ¸ 1 > > : ¡1 if f ·¡1 And 8 > > g if ¡1 · g · 1 <> P g = 1 if g ¸ 1 > > : ¡1 if g ·¡1 It is easy to see that jjP f ¡ P gjj · jjf ¡ gjj.
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