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Three Non-Linear Problems on Normed Spaces A THREE NON-LINEAR PROBLEMS ON NORMED SPACES A dissertation submitted to Kent State University in partial fulfillment of the requirements for the degree of Doctor of Philosophy by Francisco J. Garc´ıa February, 2007 Dissertation written by Francisco J. Garc´ıa B.S., University of C´adiz, Spain, 2000 M.S., University of C´adiz, Spain, 2004 Ph.D., University of C´adiz, Spain, 2005 M.A., Kent State University, 2006 Ph.D., Kent State University, 2007 Approved by Richard M. Aron, Chair, Doctoral Dissertation Committee Andrew Tonge, Members, Doctoral Dissertation Committee Per H. Enflo, Johnnie W. Baker, (Outside Person) Paul S. Wang, (Graduate Representative) Accepted by Andrew Tonge, Chair, Department of Mathematical Sciences Jerry Feezel, Dean, College of Arts and Sciences ii TABLE OF CONTENTS ACKNOWLEDGEMENTS .............................. v INTRODUCTION ................................... 1 1 THE LINEABILITY PROBLEM FOR FUNCTIONALS ......... 7 1.1 Preliminaries . 7 1.2 Lineability of NA (X)............................... 16 1.3 Lineability of X∗ \ NA (X)............................ 21 1.4 Density of X∗ \ NA (X)............................. 24 2 THE MINIMUM-NORM PROBLEM FOR TRANSLATIONS ..... 31 2.1 Preliminaries . 31 2.2 Minimum-norm elements and norm-attaining functionals . 42 2.3 Non-complete normed spaces having only norm-attaining functionals . 45 2.4 Partial solutions . 48 3 THE BANACH-MAZUR CONJECTURE FOR ROTATIONS ...... 51 3.1 Preliminaries . 51 iii 3.2 Geometrical conditions . 58 3.3 Topological conditions . 60 3.4 Intermediate solutions . 63 BIBLIOGRAPHY . 68 iv ACKNOWLEDGEMENTS There are several people to whom I should be giving thanks for my stay at Kent State University: Richard Aron, Joe Diestel, Andrew Tonge, Artem Zvavitch, Per Enflo, and Juan Seoane. I would like to use this opportunity to thank all of them very much: Thanks to Richard for accepting me as his student and for all the invitations to his house; thanks to Joe and Artem for the recommendation letters that they wrote for me; thanks to Andrew for all the forms that he signed for me; thanks to Per for his excellent mathematical support; and, finally, thanks very much Juan for making my stay at Kent as good as possible. Also, I want to give thanks to the Department of Mathematical Sciences at Kent State University for the privilege of having been supported by the Graduate Program. Specially, I wish to thank Virginia Wright (Secretary of the Math Department,) Misty Tackett (Sec- retary of the Math Graduate Students,) and Michelle Cordier (Secretary Assistant,) for attending me so well every time I needed their help. Other people I would like to thank are my friends in the Math Department like Mienie, Ramiro, Tom, Hongcheng, Jeff, Terry, Brian, Antonia, Daniele, and Alejandro. Thanks to all of you guys, overall for the rides from and to Cleveland Hopkins International Airport. Finally I mention all my friends and my family from my town. Brothers and sisters, thanks for always being there for me. Please, never change! Francisco J. Garc´ıa Febraury 2007 v GARC´IA, FRANCISCO J., Ph.D., February, 2007 PURE MATHEMATICS THREE NON-LINEAR PROBLEMS ON NORMED SPACES (71 pp.) Director of Dissertation: Richard M. Aron In this dissertation, we will study the following three non-linear problems: 1. The lineability problem for functionals. 2. The minimum-norm problem for translations. 3. The Banach-Mazur conjecture for rotations. As far as we know, all of them are currently open, and we believe that any approach to their solutions will constitute a work of great interest to the mathematical community. In this dissertation, we obtain progresses that lead to partial solutions of these problems. INTRODUCTION The first chapter of this dissertation is on the structure of the set of functionals on a Ba- nach space that attain their norm. The reason for this interest comes from an open problem concerning the lineability of the set NA (X) of norm-attaining functionals on a Banach space X. Specifically, it is unknown if NA (X) always contains an infinite dimensional, or even a 2-dimensional subspace. The paper [9] is an excellent reference about this problem. As we will see, from this problem arises another one corresponding to the complementary set, X∗ \ NA (X) , of non-norm-attaining functionals on X. In concrete terms, for non-reflexive spaces X, is X∗ \ NA (X) always lineable, or dense? Our best results from this chapter are the following: 1. Every Banach space admitting an infinite dimensional separable quotient can be equiv- alently renormed to make the set of norm-attaining functionals lineable. 2. There exists a non-reflexive dual Banach space that cannot be equivalently dually renormed to make the set of non-norm-attaining functionals even 2-lineable. 3. Every Banach space can be equivalently renormed to make the set of non-norm- attaining functionals non-dense. All the results presented in this chapter, unless explicitly stated, can be found in [2]. The second chapter consists of the study of norm-attaining functionals on non-complete spaces. In 1964, James proved that a Banach space is reflexive if and only if every func- tional is norm-attaining (see [28].) Besides, James showed in 1971 that the completeness hypothesis cannot be skipped, since he found a non-complete normed space on which every 1 2 functional is norm-attaining (see [29].) Afterwards, Blatter proved in 1976 that a necessary and sufficient condition for a normed space to be reflexive is that every closed convex set has a minimum-norm element (see [17].) This was the first time when minimum-norm elements appeared in the literature. However, they appeared again in 2005, when Aizpuru and the author generalized Blatter’s result by proving that a normed space is reflexive if and only if every bounded closed convex set with non-empty interior has a minimum-norm element (see [4].) Here, the bounded closed convex sets with non-empty interior became important, and with this we have the following conjecture: A necessary and sufficient condition for a normed space to have only norm-attaining functionals is that every bounded closed convex set with non-empty interior can be translated to have a non-zero minimum-norm element. Another interesting problem arising from this is to characterize the reflexive spaces that contain a proper dense subspace on which every functional is norm-attaining. Our best results from this chapter are the following: 1. A necessary and sufficient condition for a normed space to have only norm-attaining functionals is that every closed convex subset with non-empty interior and non-empty boundary can be translated to have a non-zero minimum-norm element. 2. If a non-complete norm space is so that every bounded closed convex subset of it with non-empty interior can be translated to have a non-zero minimum-norm element, then its completion cannot be rotund. 3. Every infinite dimensional reflexive Banach space can be equivalently renormed to be non-rotund and to not possess dense proper subspaces on which every functional is norm-attaining. All the results presented in this chapter, unless explicitly stated, can be found in [24]. 3 The third and last chapter is about a famous old problem. It has been always well known that every Hilbert space is transitive. On the other hand, it seems likely that Banach already knew some examples of transitive Banach spaces that were not Hilbert. Apparently, Mazur, who was working with separable spaces at that time, asked Banach for the existence of transitive and separable Banach spaces different from `2. Banach came up with no answer and this is how the Banach-Mazur conjecture was born: every transitive and separable Banach space is Hilbert. It is believed that Mazur conjectured a positive answer. An excellent reference is [12], and another good one is [18]. Basically, there are two geometrical properties clearly differentiated: Rotundity and smoothness. Therefore, it is natural to wonder whether a transitive and separable Banach space is rotund and smooth. In 1932 (see [33]) Mazur proved that in every separable Banach space the set of smooth points of the unit ball is a Gδ dense subset of the unit sphere. As a consequence, every transitive and separable Banach space is smooth. Now, the question remains of whether a transitive and separable Banach space is rotund. Our best results from this chapter are the following: 1. If the unit ball of a smooth and separable Banach space is free of rotund points, then the set of non-norm-attaining functionals on it contains a Gδ dense subset. 2. If a transitive and separable Banach space is so that the set of non-norm-attaining functionals on it is not dense, then the space is rotund, the set of rotund points of the unit ball of its dual is dense in the unit sphere of its dual, and the set of norm-attaining functionals on it is open. 3. If the unit ball of a transitive and separable Banach space either has normal structure or is dentable, then the space is rotund. All the results presented in this chapter, unless explicitly stated, can be found in [5]. 4 As the reader may suppose, in much of this dissertation we will make use of various classical notions, such as smoothness, rotundity, etc., from the geometry of Banach spaces. Now, we will briefly describe the notation we have followed through the whole dissertation. Assume that X denotes a topological vector space. Then: 1. If X is metrizable then BX (x, δ), UX (x, δ), and SX (x, δ) will denote the closed ball of center x and radius δ, the open ball of center x and radius δ, and the sphere of center x and radius δ, respectively.
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