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Linear and Algebraic Structures in Function Sequence Spaces

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UNIVERSIDAD DE SEVILLA Facultad de Matem´aticas Departamento de An´alisisMatem´atico LINEAR AND ALGEBRAIC STRUCTURES IN FUNCTION SEQUENCE SPACES Memoria presentada por Pablo Jos´eGerlach Mena para optar al grado de Doctor por la Universidad de Sevilla. Ma del Carmen Calder´onMoreno Jos´eAntonio Prado Bassas Prof. Titular Universidad Prof. Titular Universidad Dpto. An´alisis Matem´atico Dpto. An´alisisMatem´atico Universidad de Sevilla Universidad de Sevilla V◦ B◦ de la Directora V◦ B◦ del Director Sevilla, Febrero de 2020 Contents Resumen vii Abstract ix Introduction xi 1 Preliminaries 1 1.1 Basic concepts and definitions . .1 1.2 Some classical spaces . .3 1.2.1 Continuous functions . .3 1.2.2 Sequence spaces . .4 1.2.3 The space L0(X; µ).........................6 1.2.4 Lp(X; µ) spaces of functions for 1 ≤ p < +1 ...........7 1.2.5 L1(X; µ) space of functions . .7 1.3 Topology of sequence spaces . .9 i Contents Lin. Alg. Struc. Func. Seq. Spaces 2 Lineability 15 2.1 Introduction . 15 2.2 Main definitions . 16 2.3 On the relation between lineability, spaceability and denseness . 18 3 Modes of convergence 27 3.1 Definitions and relations between convergences . 27 3.2 Counterexamples on the reciprocals . 32 4 Lineability and modes of convergence 41 4.1 Measure vs. Pointwise a.e. Convergence . 42 4.1.1 Algebrability and spaceability . 48 4.2 Pointwise vs. Uniform Convergence . 54 4.2.1 Definitions and first examples . 54 4.2.2 Lineability results . 57 5 The space L1([0; +1)) 61 5.1 Convergence in L1(X; µ).......................... 62 5.1.1 Measure vs. L1 convergence . 62 5.1.2 Uniform vs L1 convergence . 63 5.2 Unbounded functions in CL1([0; +1)) . 71 5.3 Sequences of continuous, unbounded and integrable functions . 82 ii Lin. Alg. Struc. Func. Seq. Spaces Contents 5.4 Final remarks . 87 5.4.1 Maximal possible convergence . 87 5.4.2 On the growth . 89 5.4.3 Smoothness . 93 6 Anti M-Weierstrass sequences of functions 97 6.1 Concepts and examples . 97 6.2 Lineability within AMW ......................... 104 6.3 Algebrability within AMW ........................ 107 Bibliography 113 iii Contents Lin. Alg. Struc. Func. Seq. Spaces iv Agradecimientos A mis directores, Mar´ıa del Carmen Calder´onMoreno y Jos´eAntonio Prado Bassas, sin los cuales esta Tesis no habr´ıasido posible. Gracias por confiar en m´ı durante todos estos a~nosde carrera, m´astery doctorado. Gracias por toda la ayuda acad´emicadurante la realizaci´ony escritura de la Tesis, as´ıcomo por todos los conse- jos que me hab´eisdado a nivel personal durante todo el proceso. Os aseguro que no caen en saco roto, y espero ser un poco mejor cada d´ıagracias a vosotros. A los profesores del Proyecto ESTALMAT, por introducirme en el mundo de las matem´aticasa tan temprana edad, con el consiguiente reto que ello conlleva. Gracias a magnificos docentes y apasionados de las Matem´aticascomo Antonio Aranda, Antonio P´erezJim´enez,Manolo Delgado, Ram´onPiedra o Francisco Castro, entre muchos m´as, he seguido en el mundo de las Matem´aticas. En especial a Alfonso Carriazo, quien, adem´asde haber sido mi profesor en ESTALMAT, la carrera y el m´aster,ha sido un amigo inestimable a nivel personal, apoy´andomeen todo y mostrando su m´assincero inter´espor m´ı. Al profesor Luis Bernal, por atraerme al campo de An´alisisMatem´aticoen segundo de carrera, y darme la oportunidad de conocer a mis actuales directores, Carmen y Jos´e.Gracias por toda la ayuda y apoyo dado durante la realizaci´onde mi Erasmus en Berl´ın. v Agradecimientos Lin. Alg. Struc. Func. Seq. Spaces Al Departamento de An´alisisMatem´atico,por todo el apoyo brindado durante estos 4 a~nos,tanto en el ´ambito acad´emicocomo personal, as´ıcomo a todo el PAS de la Facultad y del IMUS por hacerme m´asagradables todos los a~nosque ha durado este viaje. A mis amigos de los Altos Colegios, Fran, Bel´en,Isaac, Elena, Jes´us,Mercedes, Mery, Alejandro, Jon (y muchos m´as),y a la Yonkozona, V´ıctor,Fran, Josemi, Javi, Julia y Adolfo, por aguantarme, animarme y ense~narmea ser mejor persona. Gracias a todos vosotros he seguido firme en este viaje por las Matem´aticas,aprendiendo de mis (muchos) errores, y tratando de dar lo mejor de m´ı. A mis padres, Lola y Frank, y a Javier, quien ha sido como un segundo padre para m´ı,a mis hermanos, Mauro, Violeta y Mateo, as´ıcomo a mi familia \iliturgitana", Isi, Pepa, Juanma y el peque~noNacho, y mi familia \bejarana", Lola, Juli´any Blanca, por apoyarme en todas mis decisiones, y darme el apoyo necesario para seguir adelante pese a todos los contratiempos sufridos, los agobios, las noches de estudio, las indecisiones y dem´as.Gracias por estar siempre a mi lado. A Teresa, mi pareja, y uno de los pilares fundamentales en todo este proceso junto a mi familia. Desde primero de carrera has estado a mi lado, vi´endomecrecer y madurar como persona. T´ume has hecho mucho mejor de lo que era, y has sufrido estos a~nosde doctorado conmigo de la mano. Todos las tardes de estudio en casa, los agobios por los ex´amenesy las notas, la alegr´ıade los premios y, por fin, el doctorado, tienen su recompensa, y buena parte de ella es gracias a t´ı. Por ´ultimo, a mi Lela, quien, a pesar de haberse ido antes de poder ver este proceso terminado, tuvo el tiempo necesario para ense~narmeque, en la vida, lo m´as importante que existe al final es el amor hacia tus seres queridos. vi Resumen Hist´oricamente han sido muchos los matem´aticosde todas las ´epocas que se han sentido atra´ıdos y fascinados por la existencia de grandes estructuras algebraicas que satisfacen ciertas propiedades que, a priori, pueden contradecir a la intuici´on matem´atica. El objetivo de la presente Memoria es el estudio de la lineabilidad de diversas familias de sucesiones de funciones con propiedades muy espec´ıficas. La Memoria se divide en 6 cap´ıtulos, donde los Cap´ıtulos1, 2 y 3 se centran en introducir la notaci´onb´asicay la terminolog´ıaprincipal de la teor´ıade la Lineabilidad y de los modos de convergencia que usaremos a lo largo de esta Memoria. En el Cap´ıtulo4 comenzamos el estudio del tama~noalgebraico de dos familias de sucesiones de funciones con distintos modos de convergencia en el intervalo unidad cerrado [0; 1]: convergencia en medida pero no puntual en casi todo y convergencia puntual pero no uniforme. En el Cap´ıtulo5 centramos nuestra atenci´onen el marco de las funciones inte- grables (Lebesgue). Comenzamos con sucesiones de funciones integrables y distintos modos de convergencia en comparaci´oncon la convergencia en norma L1, y finalizamos el cap´ıtulocon el tama~noalgebraico de las familias de funciones no acotadas, continuas e integrables en [0; +1), y las sucesiones de ellas. vii Resumen Lin. Alg. Struc. Func. Seq. Spaces Finalmente, en el Cap´ıtulo6 trabajamos en el ´ambito de las series de funciones, obteniendo resultados positivos sobre el tama~nolineal y algebraico de la familia de sucesiones de funciones cuya serie asociada converge uniformemente pero no verifica las hip´otesisdel Criterio M de Weierstrass. viii Abstract Historically, many mathematicians of all ages have been attracted and fascinated by the existence of large algebraic structures that satisfy certain properties that, a priori, contradict the mathematical intuition. The aim of the present Dissertation is the study of the lineability of certain families of sequences of functions with very specific properties. The Dissertation is divided in 6 chapters, where Chapters 1, 2 and 3 focus on introducing the basic notation and main terminology of the theory of Lineability and modes of convergence that will be used along this Dissertation. In Chapter 4 we begin with the study of the algebraic size of two families of sequences of functions with different modes of convergence in the closed unit interval [0; 1]: convergence in measure but pointwise almost everywhere and pointwise but not uniform convergence. In Chapter 5 we focus our attention on the setting of (Lebesgue) integrable func- tions. We start with sequences of integrable functions with different modes of conver- gence in comparison to the L1-convergence, and finish the chapter with the algebraic size of the family of unbounded, continuous and integrable functions on [0; +1) and sequences of them. ix Abstract Lin. Alg. Struc. Func. Seq. Spaces Finally, in Chapter 6 we turn into the setting of series of functions, obtaining pos- itive results about the linear and algebraic size of the family of sequences of functions whose series converges absolutely and uniformly but does not verify the hypothesis of the Weierstrass M-test. x Introduction In the field of Functional Analysis, one of the branches that has experienced a wider development in the past few decades is the theory of Lineability. In this branch we can find the study of the existence of vector spaces inside spaces with a strong non-linear setting, as well as the existence of linear algebras. This is precisely the framework in which the present Dissertation is set. The theory of Lineability started in 1966 with a paper of the Russian mathemati- cian Vladimir I. Gurariy [30], where he proved that the family of continuous functions in the closed unit interval [0; 1] that are nowhere differentiable contains, except for the null function, an infinite dimensional vector space. The concept of lineability (ex- istence of an infinite dimensional vector space) was coined by Gurariy himself.
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