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Descarregar (1.400Mb) On the Bishop-Phelps-Bollobás type theorems Sheldon M. Gil Dantas Supervised by Domingo García, Manuel Maestre and Miguel Martín Facultat de Ciències Matemàtiques Universitat de València This dissertation is submitted for the degree of Doctor in Mathematics (Valencia University) March 2017 I declare that this dissertation titled On the Bishop-Phelps-Bollobás type theorems and the work presented in it are my own. I confirm that: • This work was done wholly or mainly while in candidature for a degree of Doctor in Mathematics at Valencia University. • Where I have consulted the published works of others, this is always clearly attributed. • Where I have quoted from the works of others, the source is always given. With the exception of such quotations, this dissertation is entirely my own work. • I have acknowledged all main sources of help. Valencia, 26 March 2017 Sheldon M. Gil Dantas We declare that this dissertation presented by Sheldon M. Gil Dantas titled On the Bishop-Phelps-Bollobás type theorems has been done under our supervision at Valencia University. We also state that this work corresponds to the thesis project approved by this institution and it satisfies all the requisites to obtain the degree of Doctor in Mathematics. Valencia, 26 March 2017 Domingo García Manuel Maestre Miguel Martín Acknowledgements I think that everybody knows how hard it can be to write a thesis and to me it was not different at all. I chose to do it far from theplaces that I know and far from the people that I care about, but the great thing about taking the hard way is that you end up in places that you have never been before and meeting people that leave a mark on you forever. That is when you grow and the world grows for you. In the next lines, I would like to thank those people. I thank my supervisors because working with the people who love their jobs is amazing. It inspires and it motives you to go on and on every day. To Domingo García, who treated me like a real son from the very first moment since he welcomed me at the airport, helped meto get over personal problems during the first year and staying by my side all the time. To Manuel Maestre, who taught me so many things that I have already lost count. From him, I learned that one has to break a lot of stones to get real and significant things. From him, I learned that one cannot give up so easily and has to bite a problem as a dog bites the flesh and never drops it. From him, I learned that effort, dedication, and passion walk together, and without these three elements I never would have been able to finish this part of my life. To Miguel Martín, who received me in Granada for four months and made my stay a very pleasurable time. Despite the distance, we could work together in what I considered very interesting mathematical problems proposed by him. viii To Sun Kwang Kim and Han Ju Lee, for the great hospitality in South Korea. There were undoubtedly enjoyable times. It is always excellent when you find great people to work together with. To Yun Sung Choi, for the stay in Pohang and for all the support after that. To Richard Aron, who always supported me since our first meeting in São Paulo five years ago. To all members from the Departamento de Análisis Matemático of Valencia University and from the Instituto Universitario de Matemática Pura y Aplicada (IUMPA) of Polytechnic Valencia University. Especially to José Bonet, Carmina Fernández, Antonio Galbis, Jesús García Falset, Rocío Gil, Josep Martínez, Vicente Montesinos, Salvador Moll, Aníbal Moltó and Pilar Rueda. To Juan Monterde who can solve any bureaucratic problem in a simple way and who is always proud of the PhD students. To Daniela Vieira, for the initial incentive and for encouraging me to do my thesis in Valencia. To Omar Muñiz, who told me not to worry and to work really hard. To Angela Rugiano, who took so many Saturdays preparing very special dinners for us. To both of them for sharing with me such good memories in our office. To all old and current PhD students from Granada, São Paulo and Valencia Universities. Special thanks to Benjamín Alarcón, Thais Ávila, María Calvo, Juan Francisco Correcher, Karen Flórez and Alfonso Fernández, Guilherme Galina, Marline Ilha, Adilson Konrad, Danielle Mingatos, Lourdes Moreno, Darllan Pinto, Francesca Spagnuolo and Tony Wauters. To Enric Cosme, for trusting in the new generation the organization of Seminari Predocs during the last two years and for always trying to ix do the difficult task of gathering all the students together as wewerea big family. To the friends that contributed directly to this: to María José Beltrán, who taught me to be stronger and more persistent; to Carlos and Cuca Navarro, for all the deep talks through Turia river; to Elizabeth Gomes who walks, suffers and smiles with me although there is an ocean between us; to Jaime Castillo, my academic-brother, for sharing with me the office, the organization of Primer Congrés Predocs and Seminari Predocs, as well as the beers, the table games and his life-experiences; to Jorge Ros and Sandra Bombillar, for spending with me such great times in our apartment; to Vinicius Martins for the very long friendship; to Lydia Moragón for always giving me such great advice in both simple and complex situations; to Danilo Alvares and Jéssica Pavani, for always being close and trying to help me in everything they could; to Javier Marco and Nayana Dell’Armelina, for being so great together and separately and to Sofia Couto, for being such a trustworthy and loyal friend. To Marta Latorre Balado, who made me realize that simple choices generate great moments: todo es mejor contigo. To the Latorre Balado family, who made me feel at home in every single moment. To my family: to J. Carlos Dantas and Fátima Gil, because without their education, support, comprehension, motivation and persistence, it would be impossible that I was able to writing these words; to Wellington Dantas who taught me science when I was a child. Maybe it is his fault that I study mathematics nowdays. To Yuri and Eric Gil Dantas, who know what is to work far from home but also know what it feels like when we are together again. To CAPES, for the financial support. Resum Aquesta tesis està dedicada a estudiar la propietat de Bishop-Phelps- Bollobás en diferents contextos. En el Capítol1, introduïm la notació utilitzada al llarg de tota la tesis i donem un resum històric d’aquesta propietat, explicant, entre altres, els teoremes de Bishop-Phelps i Bishop-Phelps-Bollobás. El primer teorema de Bishop-Phelps [14] ens diu el següent: Siga X un espai de Banach. Donat un número real positiu ε > 0 i un ∗ ∗ ∗ ∗ funcional lineal i continu x ∈ X , existeix un nou funcional x0 ∈ X i un nou punt x0 ∈ SX tals que satisfan les següents condicions: ∗ ∗ ∗ ∗ |x0(x0)| = ∥x0∥ i ∥x0 − x ∥ < ε. Es a dir, el teorema prova que el conjunt de tots els funcionals que alcança la norma és dens (en norma) en el dual d’un espai de Banach X. És natural doncs preguntar-se si aquest resultat és també cert per a operadors lineals i continus. En 1963, J. Lindenstrauss [52] va donar el primer contraexemple que mostra que, en general, el resultat és fals. D’altra banda, també va presentar condicions que han de satisfer els espais de Banach per a obtindre resultats positius. Després d’això, altres autors han estudiat diverses hipòtesis que es poden exigir als espais de Banach per a saber si el conjunt dels operadors lineals i continus xii que alcancen la norma és dens (en norma) dins del conjunt de tots els operadors lineals i continus. Set anys més tard, en 1970, Bollobás [15] va provar una versió més forta del teorema de Bishop-Phelps, enunciada a continuació. ∗ Siga X un espai de Banach. Donat ε > 0, si x ∈ BX i x ∈ BX∗ complixen ε2 |1 − x∗(x)| < , 4 ∗ aleshores existeixen elements x0 ∈ SX i x0 ∈ SX∗ tals que ∗ ∗ ∗ |x0(x0)| = 1, ∥x0 − x∥ < ε i ∥x0 − x ∥ < ε. Amb aquest resultat Bollobás va mostrar que a més de l’aproximació per als funcionals, també és possible aproximar el punt inicial on el primer funcional x∗ quasi alcança la norma, per un punt en el que el nou funcional, proper a x∗, alcança la norma. En els últims nou anys, s’han escrit una gran quantitat d’articles en els que es presenten diferents teoremes del tipus Bishop-Phelps-Bollobás per a operadors lineals i continus. L’interès per aquests teoremes va començar amb l’article de M. Acosta, R. Aron, D. García i M. Maestre [2]. En aquest article, els autors van definir per primera vegada la propietat de Bishop-Phelps-Bollobás (BPBp). La propietat és la següent. Diem que un parell d’espais de Banach (X; Y ) satisfà la BPBp si donat ε > 0, existeix η(ε) > 0 tal que per a tot T : X −→ Y operador lineal de norma u i per a tot x0 element en l’esfera unitat de X que complixen ∥T (x0)∥ > 1 − η(ε), existeixen un nou operador lineal S : X −→ Y de norma u i un nou element x1 en l’esfera unitat de X tals que complixen les següents condi- xiii cions: ∥S(x1)∥ = 1, ∥x1 − x0∥ < ε i ∥S − T ∥ < ε. És important destacar que del teorema de Bishop-Phelps-Bollobás es dedueix que el parell (X; K) satisfà la BPBp per a tots els espais de Banach X.
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