NON-LINEAR MAPS BETWEEN SUBSETS OF BANACH SPACES

Transcription

1 NON-LINEAR MAPS BETWEEN SUBSETS OF BANACH SPACES A dissertatio submitted to Ket State Uiversity i partial fulfillmet of the reuiremets for the degree of Doctor of Philosophy by Reema Sbeih December, 009

2 Dissertatio writte by Reema Sbeih BS, Birzeit Uiversity, West Bak, Palestie 000 MS, Yougstow State Uiversity, 00 MA, Ket State Uiversity, 008 PhD, Ket State Uiversity, 009 Approved by Dr Per Eflo Dr Adrew Toge Dr Morley Davidso, Chair, Doctoral Dissertatio Committee, Member, Doctoral Dissertatio Committee, Member, Doctoral Dissertatio Committee Dr Keeth Batcher, Member, Outside Disciplie Dr Peter Tady, Member, Graduate Represetative Accepted by Dr Adrew Toge, Chair, Departmet of Mathematical Scieces Dr Joh R D Stalvey, Dea, College of Arts ad Scieces ii

3 TABLE OF CONTENTS ACKNOWLEDGEMENTS iv INTRODUCTION BASIC DEFINITIONS AND NOTATION 3 3 PROJECTIONS ONTO THE UNIT BALL IN L P SPACES 7 4 SCALING DOWN MAPS BETWEEN Sl AND Sl 4 5 SCALING DOWN MAPS BETWEEN Sl AND Sl 4 6 SCALING DOWN MAPS BETWEEN Sl AND Sl as 37 7 CONCLUSION AND FUTURE PROJECTS 48 BIBLIOGRAPHY 50 iii

4 ACKNOWLEDGEMENTS I owe this work to the two people from whom I received my life s blood, my parets My ultimate ispiratio comes from my advisor, Dr Per Eflo, the woderful mathematicia who guided through this icredible ad log jourey iv

5 CHAPTER INTRODUCTION I this dissertatio we study o-liear maps betwee subsets of Baach spaces As a backgroud for this study we should metio two areas of mathematical research: Geometry of Baach Spaces ad Approximatio Theory Oe way to study ad compare the geometry of differet Baach spaces is to study liear maps betwee the spaces ad to study how much these maps distort distaces betwee poits There is a extesive literature o this [], [3], [4], [5] Aother less studied way to compare the geometry of differet Baach spaces is to study maps betwee differet subsets of the spaces I this case the maps are usually o-liear ad the subsets are ofte uit balls or uit spheres of Baach spaces [6], [7], [0], [] I Approximatio Theory, the followig problem is importat: how well ca objects poits from a set A be approximated by objects poits from a set B? Oe importat way to study this approximatio is to fid for every poit i A the earest poit i B, ie, to fid the earest poit map, also called the metric projectio, ad ivestigate its properties [0], [], [3] The maps that we study i this dissertatio, the scalig dow projectio ad the scalig dow maps, are, as we shall see, earest poit maps The dissertatio is orgaized as follows: I Chapter we give the basic defiitios ad otatio that we use i the rest of the paper We also show that the scalig dow projectio ad the scalig dow maps are earest poit maps I Chapter 3, we show that the scalig dow projectio oto the uit sphere i ay

6 Baach space has Lipschitz orm at most ad that is ot attaied We also show that i L 0,, the scalig dow projectio has orm, ad that the scalig dow projectio i L 0, is the best projectio i the sese that all other projectios have orm We also give estimates for the orm of the scalig dow projectio i L p for < p < I chapters 4, 5, ad 6 we study scalig dow maps betwee the uit sphere of l, beig i some sese the largest uit sphere, ad the uit spheres of l-spaces p I particular we study the Baach-Mazur orms of these maps Our results show that the Baach-Mazur orms of these maps are much larger tha the Baach-Mazur orms of the simplest liear maps betwee the spaces

7 CHAPTER BASIC DEFINITIONS AND NOTATION I this chapter we give the basic defiitios ad otatio that we use i this paper We also show that the scalig dow projectio ad the scalig dow map are earest poit maps Defiitio [7] A vector space X is said to be a ormed space if for every x X there is associated a oegative real umber x, called the orm of x, i such a way that a x + y x + y for all x ad y i X, b αx α x if x X ad α is a scalar, c x > 0 if x 0 Every ormed space maybe regarded as a metric space, i which the distace dx, y betwee x ad y is x y Defiitio [7] A Baach space is a ormed space which is complete i the metric defied by its orm; this meas that every Cauchy seuece is reuired to coverge Let X, d be a ormed space Defiitio The uit ball of X, deoted by BX is BX {x X : dx, 0 } Defiitio The uit sphere of X, deoted by SX is SX {x X : dx, 0 } 3

8 4 Defiitio A projectio is a map P : X X such that P P P Defiitio We say that M X is cotractive if there is a projectio P i geeral oliear from X oto M such that dp x, P y dx, y for all x, y X Defiitio Let M, N X A map T : M N is a earest poit map if dx, T x dx, z for all z N Defiitio A map P o X is called a Lipschitz map if there is a costat C such that P x P y C x y for all x, y X The smallest costat C that satisfies is called the Lipschitz orm of P, ad P x P y is deoted by P It is easy to see that P sup x,y X x y Defiitio The map P o X defied by x if x > P x x x if x is called the scalig dow projectio Cosider the orms ad o a vector space Assume x x for every x Let X, Y be the ormed spaces associated with x ad x respectively We have the followig defiitio

9 5 Defiitio The scalig dow map T : SX SY is the map T that takes x γ x x where x ad γ x x Lemma The scalig dow projectio is a earest poit map from X oto BX Proof Let x X ad let y P x If x, the y x ad x y 0 So suppose that x > ; the y x x We eed to show that if z is ay poit i BX the x y x z We have x y x x x x x x, ad x x x z x z x Therefore x y x z for all z BX Corollary The scalig dow map ad its iverse are earest poit maps Defiitio A ormed space X is strictly covex if tx + tx < wheever x ad x are differet poits of SX ad 0 < t < Defiitio A ormed space X is reflexive if every bouded seuece has a weakly coverget subseuece Defiitio A ormed space X is smooth if every poit o the uit sphere of X has a uiue supportig hyperplae Next we will give some of the otatio used i this paper L p E {f : E f p < }, 0 < p < f p f p p, 0 < p < E f if{α : m{x E : fx > α} 0} L E {f : f < }

10 6 Let a {a k } be a seuece of real or complex umbers The we have the followig defiitios: a p k a k p p, 0 < p < a sup k a k l p {a : a p < }, 0 < p < l {a : a < }

11 CHAPTER 3 PROJECTIONS ONTO THE UNIT BALL IN L P SPACES It is kow that uit balls i Hilbert spaces ad spaces of cotiuous fuctios are cotractive I this chapter, we show that i ay Baach space, the scalig dow projectio has Lipschitz orm, ad has orm eual to i L 0, We also show that i L 0,, ay projectio other tha the scalig dow projectio has Lipschitz orm at least We also estimate the orm of the scalig dow projectio i L p 0,, < p < I a Hilbert space, a closed, covex set is cotractive sice the earest poit map is a cotractive projectio Theorem 3 [] Beauzamy-Maurey result Let X be reflexive, strictly covex, smooth Baach space of dimesio 3 or greater If the uit ball of X is cotractive the X is a Hilbert space Remark Sice l ad C0, are ot strictly covex, we caot apply the Beauzamy- Maurey result However, i C0, the uit ball is cotractive To see this, let f, g C0, 7

12 8 Let P f f if f if f if f Ad P g g if g if g if g It is easy to see that P f P g f g Let X be a ormed space Lemma 3 Let x, y X such that x + ɛ ad y Let P be the scalig dow projectio o X The P x P y < x y Proof Let z P x The x + ɛy + ɛz + ɛy + ɛ z y Also x + ɛy x y ɛy x y + ɛ So + ɛ z y x y + ɛ

13 9 Divide both sides of this ieuality by + ɛ x y We get This gives z y x y + ɛ + ɛ + ɛ x y + ɛ + + ɛ + ɛ sice x y ɛ 3 P x P y x y + ɛ Theorem 33 Let x, y X, x y, ad let P be the scalig dow projectio o X The 3 P x P y < x y Proof Let x, y X with x > ad y > Suppose wlog that x > y The 33 x y y y < x y So it is eough to prove the theorem for the case where y which is doe i Lemma 3 I the ext theorem, we show that i L 0,, the Lipschitz orm of the scalig dow projectio is Theorem 34 I L 0,, the Lipschitz orm of the scalig dow projectio is Proof From Theorem 33 we have that the Lipschitz orm To show that the Lipschitz orm is, it is eough to fid oe fuctio fx, outside the uit ball of

14 0 L 0, that caot be projected to the uit ball without gettig the Lipschitz orm arbitrarily close to Take fx + ɛ o [0, ] Let h P f, h, the h o [0, ] Defie the fuctio g as follows: + ɛ o A, for some A [0, ], ma gx + ɛ 0 o A c the g Ad This gives f g + ɛ + ɛ ɛ h g ɛ + ɛ + ɛ + ɛ + ɛ h g f g + ɛ as ɛ 0 Theorem 35 Let P be a projectio oto the uit ball i L 0, If P is ot the scalig dow projectio, the P Proof We show that for every projectio P that is ot the scalig dow projectio, there are g ad f, g, f + ɛ such that 34 P f P g f g + ɛ Cosider f + ɛ o [0, ] Let h P f, h Let B be a set of measure ɛ such that ht dt ɛ The ht dt ɛ Put B [0,] B gt 0 if t B ɛ if t [0, ] B

15 We ca see by the defiitio of g that g The g f gt ft dt + B ɛ + ɛ + ɛ [0,] B ɛ + ɛ + ɛ ɛ + ɛ gt ft dt + ɛ ɛ We have P g P f g h sice P g g ad P f h Now g h B gt ht dt + [0,] B gt ht dt We eed to fid gt ht dt ad gt ht dt We start with gt B [0,] B B ht dt We kow that gt 0 o B ad that ht dt ɛ So gt ht dt B B ɛ Now we eed to fid gt ht dt [0,] B [0,] B gt ht dt [0,] B gt dt [0,] B ht dt So This gives gt ɛ P g P f B o [0, ] B ad ɛ ɛ, sice gt ht dt + P f P g f g [0,] B [0,] B ɛ ɛ + ɛ + ɛ ht dt ɛ gt ht dt ɛ If f is ot mapped oto the uit sphere but P f < the let Qf be the poit o the lie segmet tf + tp f, 0 < t < such that Qf The, as above

16 Qf g we fid a g, g such that Obviously P g g Sice Qf is f g + ɛ o the lie segmet betwee f ad P f we obviously have Qf g max f g, P f g P f g For this case proof of the theorem is complete P f P g f g, ad the + ɛ We do ot kow if the scalig dow projectio i L p 0,, < p < has miimal Lipschitz orm The proof we gave for the case where p does ot geeralize I the ext theorem, we give a estimate from below of the orm of the scalig dow projectio i L p 0, for < p < We observe that this estimate goes to as p ad goes to as p Theorem 36 The Lipschitz orm of the scalig dow projectio i L p 0,, < p < is greater tha or eual to p p p + p p where is arbitrary such that < p < Proof Take fx + ɛ o [0, ] Let h P f, the h o [0, ] Defie the fuctio g as follows, + ɛ o A [0, ], ma p gx δ o A c where δ is such that g p We have g p which implies + ɛ p p + δ p p So we get δ ɛ p p Where we used the approximatio + ɛ p + ɛp ad δ p δp, the error

17 3 beig of order of magitude ɛ ad δ, respectively We get g f p p ɛ + ɛ p p p p, ad p g h p p ɛ p p ɛ p + p p Therefore g h p p g f p p ɛ p ɛ p p + p ɛ + ɛ p p p p p p p + p p p p p p p p + p p p p p p p p p + p p By choosig large, we get that as p, the expressio above goes to Ad by lettig, the expressio above goes to as p

18 CHAPTER 4 SCALING DOWN MAPS BETWEEN Sl AND Sl I this chapter, we study scalig dow maps betwee the uit spheres of l ad l These are, i a sese, the simplest possible maps betwee these uit spheres ad have the property that both the map ad its iverse are earest poit maps We show that the Lipschitz orms of these maps are of the order of magitude, so the Baach-Mazur orm, as defied below, is of the order of magitude This should be compared to the simplest possible liear maps where the Baach-Mazur orm is We start with some defiitios Let X, d, Y, d be metric spaces ad let M X, N Y Defiitio A map T : M N is called a Lipschitz map if there is a costat C such that 4 d T x, T y Cd x, y for all x, y M The smallest costat C that satisfies 4 is called the Lipschitz orm of T, ad is deoted by T It s easy to see that T sup x,y M d T x, T y d x, y Defiitio If T : M N is a oe-to-oe ad oto Lipschitz map, ad T is also Lipschitz, we defie the Baach-Mazur orm of T as T T 4

19 5 Theorem 4 If T is the scalig dow map from Sl to Sl the T d T x, T y Proof We eed to show that T sup, where x ad y are poits x,y d x, y o the uit sphere of l Let d deote the distace i l ad d deote the distace i l Take x, 0 ad y 0, The d x, y ad d T x, T y d x, y Which implies T To show that T, we show that d T x, T y d x, y 0, There are four cases to cosider: wheever x, 0 ad y x, a, y b,, 0 a, b, a b x, a, y b,, 0 a, b 3 x, a, y b,, 0 a, b, a b 4 x, a, y, b, 0 a, b, a b We show that if x ad y satisfy ay of the four cases above, the d T x, T y d x, y Case x, a, y b,, 0 a, b, a b This gives Thus d T x, T y d x, y a T x, a + a + a, a + a T y b b, + b + b, + b + a b + + b + b a + a a + a + b + b ab + a + b

20 6 So d T x, T y d x, y ab a + b Case x, a, y b,, 0 a, b This gives Thus So d x, y + a T x, a + a T y b, + b d T x, T y + a, a + a b + b, + b + a b + b + + b + a + a + a + a + b + b + b d T x, T y d x, y + a + b Case 3 x, a, y b,, 0 a, b, a b This gives Thus d x, y + b T x, a + a T y b, + b d T x, T y + a, a + a b + b, + b + a + b + b + + b + a + a

21 7 So d T x, T y d x, y + b Case 4 x, a, y, b, 0 a, b, a b This gives Thus T x, a + a T y, b + b d T x, T y d x, y a b + a, a + a + b, b + b + a + b + b + b a + a a b + a b a b So d T x, T y d x, y Cases through 4 show that T This proves Theorem 4 Theorem 4 If T is the scalig dow map from Sl to Sl, 3, the T Proof We start by showig that T Take x,, ad y, a,, a,, a a, where a is a large positive umber This gives d x, y a

22 8 T x T y a, a + a a +, a,, a,, a a So Thus d T x, T y a a + d T x, T y d x, y a + a a + a a + a as a a + So T sup x,y d T x, T y d x, y Now we eed to show that T To do that, take two poits x ad y o the uit sphere of l Let x a, a,, a, a i for i,, ad a i for some i, ad y b, b,, b, b i for i,, ad b i for some i Here d x, y max a i b i i Let a i A ad b i B, the i i a T x A, a A, a A b T y B, b B, b B Thus d T x, T y a A b B + a A b B + + a A b B a i A b i B i

23 9 Notice that So Now a i A b i B a i A b i B i a i B b i A AB a i B b i B + b i B b i A AB a i B b i B + b i B A AB a i b i A i i i B A a i b i A a i b i A a i b i A + b ib A AB i i b i B A AB b i B A AB B A A b i a i i bi a i i Claim: b i a i a i b i Proof a i b i a i b i ad b i a i b i a i Therefore a i b i a i b i So B A This gives a i b i i d T x, T y a i A b i B i a i b i A i max i a i b i

24 0 So d T x, T y d x, y This gives T Theorem 43 If T is the scalig dow map from Sl to Sl,, the T Proof We start by showig that T To do this, we show that if x,y sphere of l To show that if x,y the uit sphere of l satisfyig d T x, T y d x, y Take x,,, d T x, T y where x ad y are poit o the uit d x, y d T x, T y, we eed to fid poits x ad y o d x, y ad y, δ, δ,, δ, where δ i 0, i,, Assume δ i < δ < δ i i, i This gives d x, y δ T x,,,, Let k δ i i T y, δ,, δ δ i i

25 The d T x, T y k + δ k δ + + k k + + δ k k i δ i Notice that Now So We get i k δ i k + + k i k This gives d T x, T y k + k + i δ i i δ i + i δ i δ i d T x, T y δ i + i + d T x, T y d x, y i δ i + δ + δ δ δ + + for large

26 So if x,y d T x, T y d x, y This gives T To show that T, take two poits x, x o the uit sphere of l such that x x a, where 0 < a < Let T x y ad T x y We kow that x is o the uit sphere of l, so x, which implies that x Now y c x for some costat c, which implies y c x, we get c x, which implies c sice x Similarly y c x with c Suppose c c ad cosider the poits z c y ad z c y By Theorem 33: x x, which implies z z a z z sice x x a But z z y y y y c c c Thus y y c z z a c So y y a c So y y x x c sice c This gives T Theorem 44 The Baach-Mazur orm of the scalig dow map from Sl to Sl is greater tha or eual to 4 ad less tha or eual to 8 Proof Let T be the scalig dow map from Sl to Sl I Theorem4, we showed that T I Theorem43, we showed that T 4 So 4 T T 8

27 3 Theorem 45 The Baach-Mazur orm of the scalig dow map from Sl to Sl is greater tha or eual to ad less tha or eual to 4 Proof Let T be the scalig dow projectio from Sl to Sl I Theorem4, we showed that T I Theorem43, we showed that T So T T 4

28 CHAPTER 5 SCALING DOWN MAPS BETWEEN Sl AND Sl I this chapter, we do a similar study as i chapter four, but comparig istead l ad l Theorem 5 If T is the scalig dow map from Sl to Sl, the T d T x, T y Proof We eed to show that T sup, where x ad y are x,y d x, y poits o the uit sphere of l Here, d deotes the distace i l ad d deote the distace i l Take x, 0 ad y 0, The d x, y ad d T x, T y d x, y So d T x, T y This gives T To show that T d x, y d T x, T y, we show that wheever x, 0 ad y 0, There are d x, y five cases to cosider: x, a, y b,, 0 a, b, a b x, a, y b,, 0 a, b 3 x, a, y b,, 0 a, b, a b 4 x, a, y, b, 0 a, 0 b, a b 5 x, a, y, b, 0 a, 0 b, a b Case x, a, y b,, 0 a, b, a b This gives d x, y a T x, a + a 4

29 5 Thus d T x, T y T y b, + b b + a + a + b + b + a + a b + a + b + b + b + + b a + a + b + a + a a + b + a + b a + b + a + b So d T x, T y a + b + a + b To show that d T x, T y, we eed to show that d x, y a + b a for 0 a, b, a b Notice that + a + b a + b + a + b is largest whe a b So a + b + a + b a + a a + a a So d T x, T y d x, y a a

30 6 Case x, a, y b,, 0 a, b This gives T x d x, y + a T y, a + a b, + b Thus d T x, T y b + + a + a + b + b + a + a b + a + b + b + b + + b a + + a + b + a + a a b + + a + b a b + + a + b So d T x, T y + a b + a + b To show that d T x, T y, we eed to show that d x, y a b + + a for 0 a, b + a + b Notice that So + + a b + a + b is largest whe b 0 a b + a + b a + + a + a

31 7 Thus d T x, T y d x, y + a + a Case 3 x, a, y b,, 0 a, b, a b This gives T x T y d x, y + b, a + a b, + b So d T x, T y + b + + a + a + b + b + a + a + b + a + b + b + b + + b a + + a + b + a + a a + b + + a + b a + b + + a + b So d T x, T y + a + b + a + b To show that d T x, T y, we eed to show that d x, y a + b + + a + b + b Notice that + a + b + a + b is largest whe a b

32 8 So + a + b + a + b + b + b + b + b + b Thus d T x, T y d x, y Case 4 x, a, y, b, 0 a, 0 b, a b This gives d x, y a + b T x T y, a + a, b + b So d T x, T y a + + b + a + b + a + b ab + a + b ab + a + b To show that d T x, T y d x, y, we eed to show that ab + a + b a + b Notice that ab + a + b ab + b sice a b b + ab + b ba + b + b a + b

33 9 Thus d T x, T y d x, y Case 5 x, a, y, b, 0 a, 0 b, a b This gives d x, y b a T x T y, a + a, b + b So d T x, T y a + b + a + b + a + b + ab + a + b + ab + a + b To show that d T x, T y + ab, we eed to show that d x, y + a + b b a + ab + a + b b a b a + ab + a + b b a + b a 4 + ab + a b + a + b b a + b a 4 + a + b + ab + a b + a + b + a b b a + b a 4 + ab + a b + a + b + a b b a + a + b + a b + b a 4 + a + b + a b + ab + a b a ab + b b a + a + b + a b b a 4 + a + b + a b Here, a ab + b b a, divide both sides of the last ieuality by b a, we

34 30 get + a + b + a b b a + a + b + a b b a + a + b + a b This ieuality holds which implies that the ieuality + ab + a + b b a also holds So d T x, T y d x, y Cases through 5 show that T This proves Theorem 5 Theorem 5 If T is the scalig dow map from Sl to Sl, 3, the T Proof We start by showig that T Take x,,,, a a ad y,,,, a a where a is a large positive umber This gives d x, y a T x, a + a T y, a + a sice x y + a a,, a,,, a, a a + a

35 3 So d T x, T y + a + a + a + a + Thus So T sup x,y d T x, T y d x, y d T x, T y d x, y a a + a a + as a Now we eed to show that T To do this, take two poits x ad y o Sl Let x a, a,, a, a i for i,, ad a i for some i, ad y b, b,, b, b i for i,, ad b i for some i Here d x, y max a i b i Let a i A ad b i B, the i i i a T x A, a A, a b ad T y A B, b B, b B [ ai Thus d T x, T y A b ] i B i [ ] ai B b i A AB i [ ] ai b i B + b i B A AB i [ ai b i A + b ] ib A AB i [ ] ai b i + b i B A, here we used the A A B i ieuality a + b a + b for ay two real umbers a ad b

36 3 [ i [ i [ i a i b i + b A i i a i b i A + a i b i A + B A A B A ] a i b i i B A b i a i i i A A B ] ], sice Take the suare root of both sides of the euality above, we get B A b i a i i i y x But y x x y a i b i i Thus B A a i b i i This gives B A So So a i b i i d T x, T y a i b i i A a i b i d T x, T y d x, y i max i a i b i max i a i b i

37 33 This gives T Theorem 53 If T is the scalig dow map from Sl to Sl,, the Proof We first show that T T d T x, T y To do this, we show that if where x ad y are poits o the x,y d x, y uit sphere of l d T x, T y To show that if, we eed to fid poits x,y d x, y x ad y o the uit sphere of l satisfyig d T x, T y d x, y Take x,,, ad y, δ,, δ, where δ 0 The d x, y δ T x,, T y, δ,, δ + δ Which gives d T x, T y + δ + δ + δ To see what d d approaches, we will use the followig approximatios: + δ + δ + δ + δ + δ δ δ δ δ,

38 34 this follows from the above approximatio for + δ Now δ δ + δ δ δ + δ δ δ, agai this follows from the above approximatio for + δ Now δ δ δ δ δ + δ δ + δ δ This gives d T x, T y δ + 3 δ + 3 δ δ δ δ

39 35 Recall that d x, y δ, which implies that d T x, T y d x, y So d T x, T y d x, y So if x,y d T x, T y d x, y This gives T To show that T, take two poits x, x o the uit sphere of l such that x x a, where 0 < a < Let T x y, ad T x y We kow that x is o the uit sphere of l, so x, which implies that x Now y c x for some costat c, which implies y c x, we get c x sice y is o the uit sphere of l Which implies c sice x Similarly y c x with c Suppose c c ad cosider the poits z y ad z y By Theorem 33, x x So z z a c c z z sice x x a But z z y y c c y y Thus y y c c z z a c So y y a c So y y c x x sice c This gives T Theorem 54 The Baach-Mazur orm of the scalig dow map from Sl to Sl is greater tha or eual to ad less tha or eual to 4 Proof Let T be the scalig dow map from Sl to Sl I Theorem5, we showed that T I Theorem53, we showed that T So T T 4

40 36 Theorem 55 The Baach-Mazur orm of the scalig dow map from Sl to Sl, 3 is greater tha or eual to ad less tha or eual to Proof Let T be the scalig dow map from Sl to Sl, 3 I Theorem5, we showed that T I Theorem53, we showed that T So T T

41 CHAPTER 6 SCALING DOWN MAPS BETWEEN Sl AND Sl as I this chapter, we do a similar study as i chapters four ad five, but comparig istead l ad l as We start with some defiitios Let X, d, Y, d be metric spaces ad let M X, N Y Defiitio Give x 0 X Defie B ɛ x 0 {x X : d x 0, x < ɛ} Defiitio A map T : M N is called locally Lipschitz at x 0 if T is Lipschitz o B ɛ x 0 for some ɛ > 0 Defiitio Give a locally Lipschitz map T : M N at x 0 Defie, T ɛ,x0 sup x,y B ɛ x 0 d T x, T y d x, y Defiitio The local Lipschitz orm at x o of a locally Lipschitz map T : M N at x 0, deoted by T loc x0, is T loc x0 lim ɛ 0 T ɛ,x0 Defiitio If T : M N is oe to oe ad oto ad locally Lipschitz at x 0, ad if T is also locally Lipschitz at T x 0, we defie the local Baach-Mazur orm of T 37

42 38 at x 0 as T loc x0 T loc T x0 Theorem 6 If T is the scalig dow map from Sl Sl,, the lim T T Proof We first prove the theorem i dimesio i order to itroduce the techiue that will be used ad the move to higher dimesios Let T : Sl Sl be the scalig dow map We show that: lim T lim T We will start with Take x, 0 ad y 0, The d x, y, ad d T x, T y d x, y d T x, T y So So T This gives lim T d x, y Take x, ad y, δ, where δ a, 0 < a < The d x, y δ T x t,, 0 < t < such that t + t This gives t So T x, Similarly T y s, δ, 0 < s < such that s + s, δ

43 39 This gives s So T y + δ, + δ Thus d T x, T y δ + δ [ ] + + δ [ δ + δ To see what d d approaches as, we will use few approximatios As oted earlier, δ a, 0 < a < ] So d T x, T y + a + a + a d We are iterested i lim d Note that: lim + a + ea e a + e a lim So d a + a e a [ e a e a + e a e a + e a + ] + [ ] e a + [ l e a e a + l ] + [ l l e a + ] Where we used the fact that l + b b, i other words lb b for small b

44 40 So d This gives d Now we use the fact that [ l e a e a + l ] [ + l ] l e a + [ ] [ ] l ea + l ea + e a + [ ] [ ] l ea + l ea + e a + We eed to fid a such that: To do that we eed to solve e a e a + ea + 4e a e a + 4e a e a + e a + e a e a + 0 e a 0 lim a + a max{a, a } l ea e a + l ea + e a e a + ea + for a a 0 So l ea e a + l ea + whe a is small So for small a, d l ea + Note that l ea + l a +, here we used the fact that ea a + So, l ea + l a + a l + a, usig the fact that lb + b for small b So d a

45 4 Thus d d a a This gives lim T Thus lim T T This proves the theorem for dimesio Next, we will prove the theorem i dimesio Let T : Sl Sl be the scalig dow map We will show that: lim T lim T We will start with Take x, 0,, 0 ad y 0,, 0, The d x, y ad d T x, T y d x, y Thus T This gives lim T Take x,, ad y, δ, δ,, δ where δ a, 0 < a < The d x, y δ T x t,,, 0 < t < such that t + + t This gives t So T x,, Similarly T y s, δ,, δ, 0 < s < such that s + s δ + + s δ [ ] This gives s + δ + + δ

46 4 Let A + δ + + δ δ δ The T y,,, A A A We eed to fid d T x, T y [ d T x, T y A + [ δ A ] + ] [ δ A ] + To see what d approaches as, we use few approximatios As oted earlier, d δ a, 0 < a < We get: lim δ lim a e a lim δ lim a e a lim A + e + + [ a e a e a So d + e a + + e a [ + + [ Let B + e a + + e a e a + e a + + e a ] e a + e a + + e a + e a + + e a ] + ] So d [ e a B ] + [ ] e a + B + [ B ]

47 43 So d [ l e a B l ] + [ l ] e a l + B + [ l l B ] Where we used the fact that lb b for small b So d [ l e a B ] [ ] [ B + l + + l B ] e a d To fid lim, we use the fact that d lim a + a + + a max i {a i} All the terms iside the parethesis of the expressio for d are approximately eual whe a is small So d l B l + ea + + e a + + a a l Where we used the fact that e b + b for small b So a d + l l + a a, usig the fact that l + b b for small b

48 44 Fially This gives Therefore d a d a lim T lim T T Defiitio A poit x Sl is a smooth poit if oe coordiate of x is ad the other coordiates have absolute value < I Theorem 6, we will prove that if we ca cosider poits that are smooth o the uit sphere of l, the locally aroud those poits we will have lim T loc T loc Theorem 6 If T : Sl Sl is the scalig dow map, the for every smooth poit x i Sl, we have lim T loc x T loc T x Proof We first prove the theorem i dimesio Let T : Sl Sl be the scalig dow map Take x, δ ad y, δ γ where 0 < δ < is fixed ad γ 0 as We show that lim T loc x T loc T x For the poits x, y, we have d x, y γ Ad T x t, δ, 0 < t < such that t + t δ This gives t + δ δ Thus T x, + δ + δ

49 45 Similarly T y s, δ γ, 0 < s < such that s + s δ γ This gives s + δ γ Thus T y + δ γ, We eed to fid a estimate for d as d We have d γ, we eed to fid d d T x, T y [ + δ [ δ + + δ δ γ + δ γ + δ γ δ γ + δ γ ] ] To see what d approaches as, we make some observatios d Note that δ 0 as sice 0 < δ < Similarly δ γ 0 as So + δ Hece [ + δ as Ad as + δ γ ] 0 as + δ γ This gives a estimate for the first term i d We eed to fid a estimate for the [ ] δ δ γ term + δ + δ γ Note that Similarly δ + δ δ γ + δ γ δ + δ δ γ δ as + δ γ δ γ as Fially, we have d [ δ δ γ] γ as

50 46 So d T x, T y γ So d T x, T y d x, y γ γ Fially lim T loc x T loc T x This proves the theorem for dimesio Next, we will prove the theorem i dimesio Let T : Sl Sl be the scalig dow map Take x, δ,, δ, ad y, δ γ,, δ γ, where 0 < δ i <, i are fixed ad γ i 0 as for i For the poits x, y, we have, d x, y max {γ i} i To fid d, we eed to fid T x ad T y T x t, δ,, δ, 0 < t < such that t + t δ + + t δ This gives t + δ + + δ Put K + δ + + δ δ δ Thus T x,,, K K K Similarly T y s, δ γ,, δ γ, 0 < s < such that s + s δ γ + + s δ γ This gives s + δ γ + + δ γ Put M + δ γ + + δ γ

51 47 Thus T y M, δ γ M,, δ γ M We eed to fid a estimate for d as We have d max d {γ i}, we eed to i fid d d T x, T y [ M K ] + [ δ K δ γ M ] + + [ δ K δ γ M ] To see what d d approaches as, we make some observatios: lim + δ + + δ sice 0 < δi < for all i lim + δ γ + + δ γ 0 < δ i, γ i < for i δi lim lim δ i lim K δi γ i lim M lim δ i γ i lim From these observatios, we get that d γ + + γ as So d [ γ + + γ ] Which implies d sice δi, i K δi γ i, i M max {γ i} as i So d T x, T y d x, y Fially lim T loc x T loc T x

52 CHAPTER 7 CONCLUSION AND FUTURE PROJECTS The ivestigatios ad results i this thesis provide a opeig for further research, which we will briefly discuss here For the projectios oto the uit ball ivestigated i Chapter three, it is atural to ask whether the assumptio that outside poits be projected oto the uit sphere ca be removed The followig example i the Euclidea plae shows that projectios which do ot map outside poits to the boudary ca have as small Lipschitz orm as those which do Example: For each α, 0 α, cosider the followig projectio oto the lower half-plae i R For y 0, P α x, y x, αy For y < 0, P α x, y x, y It is easy to see that P α, but oly for α 0 does P α map outside poits to the boudary Theorem 36 estimates the orm of the scalig dow projectio oto the uit ball for < p < It is atural to cojecture that for < p, the scalig dow projectio is the projectio oto the uit ball of L p with miimal Lipschitz orm However, the proof of Theorem 34 which gives the result for p does ot geeralize i ay simple way For < p < we do ot have a cojecture about what is the projectio oto the uit ball i L p with miimal Lipschitz orm For p, it is ot the scalig dow projectio Sice L 0, is a subspace of L ad l, theorem 34 gives that the scalig dow projectio i L ad l has orm, but the trucatio projectio has orm I Chapters four, five, ad six, the Baach-Mazur orm of the scalig dow map 48

53 49 betwee the uit spheres of l ad l is estimated to a exact order of magitude The problem about the order of magitude of the Baach-Mazur distace betwee these uit spheres remais ope For the liear case it is easy to see that the Baach- Mazur orm of the idetity map betwee l ad l is exactly But it is well kow that the Baach-Mazur distace betwee l ad l is of the order of magitude I this dissertatio we have limited the ivestigatios to the most classical Baach spaces It should be a iterestig task to study the correspodig problems for other Baach spaces as well as for other topological liear spaces like the L p -spaces where 0 < p <

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