DOCSLIB.ORG

Linear and Algebraic Structures in Function Sequence Spaces

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Linear and Algebraic Structures in Function Sequence Spaces 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.
Load more

Recommended publications
  • Functions of a Complex Variable II Math 562, Spring 2020
    Functions of a Complex Variable II Math 562, Spring 2020 Jens Lorenz April 27, 2020 Department of Mathematics and Statistics, UNM, Albuquerque, NM 87131 Contents 1 Notations 5 2 The Schwarz Lemma and Aut(D) 6 2.1 Schwarz Lemma . .6 2.2 Biholomorphic Maps and Automorphisms . .7 2.3 The Automorphism Group of D ...................7 2.4 The Schwarz{Pick Lemma . .9 2.5 Remarks . 11 3 Linear Fractional Transformations and the Riemann Sphere 12 3.1 The Riemann Sphere . 12 3.2 Linear Fractional Transformations . 14 3.3 The Automorphisms of the Riemann Sphere . 16 3.4 A Remarkable Geometric Property of Linear Fractional Trans- formations . 17 3.4.1 Analytical Description of Circles . 18 3.4.2 Circles under Reciprocation . 18 3.4.3 Straight Lines under Reciprocation . 20 3.5 Example: The Cayley Transform in the Complex Plane . 21 3.6 The Cayley Transform of a Matrix . 22 4 The Riemann Mapping Theorem 24 4.1 Statement and Overview of Proof . 24 4.2 The Theorem of Arzela{Ascoli . 26 4.3 Montel's Theorem . 29 4.4 Auxiliary Results on Logarithms and Square Roots . 32 4.5 Construction of a Map in ..................... 33 4.6 A Bound of f 0(P ) for all Ff .................. 35 4.7 Application ofj thej Theorems2 of F Montel and Hurwitz . 36 4.8 Proof That f is Onto . 37 1 4.9 Examples of Biholomorphic Mappings . 39 5 An Introduction to Schwarz–Christoffel Formulas 41 5.1 General Power Functions . 41 5.2 An Example of a Schwarz–Christoffel Map .
    [Show full text]
  • Power Series, Analyticity and Elementary Functions
    MATH 566 LECTURE NOTES 2: POWER SERIES, ANALYTICITY AND ELEMENTARY FUNCTIONS TSOGTGEREL GANTUMUR 1. Uniform and normal convergences For a set Ω ⊆ C and a function f :Ω ! C, we define the uniform norm kfkΩ = sup jfj: Ω We say that a sequence ffkg of functions fk :Ω ! C converges uniformly in Ω to f, if kfk − fkΩ ! 0 as k ! 1. Recall that this is equivalent to the sequence ffkg being a Cauchy sequence, i.e., kfk − f`kΩ ! 0 as k; ` ! 1. P P Theorem 1 (Weierstrass M-test). If gn :Ω ! C and n kgnkΩ < 1, then the series n gn converges uniformly in Ω. P Proof. With fk = n≤k gn, we have for ` < k X X X kfk − f`kΩ = k gnkΩ ≤ kgnkΩ ≤ kgnkΩ; `<n≤k `<n≤k n>` which tends to 0 when ` ! 1. So ffkg is a Cauchy sequence, hence converges. P n ¯ Example 2. The geometric series n z converges uniformly in Dr (and also in Dr) as long n as r < 1. To be pedantic, with r < 1 and with functions gn : Dr ! C given by gn(z) = z , P the series n gn converges uniformly in Dr. However, the convergence is not uniform in the P n open unit disk D1, and n z does not converge if jzj ≥ 1. Convergence behaviour of frequently occurring sequences in complex analysis can be cap- tured conveniently by the notion of locally uniform convergence. Recall that U ⊆ Ω is a neighbourhood of z 2 Ω in Ω if there is an open set V ⊆ C such that z 2 V \ Ω ⊆ U.
    [Show full text]
  • Domains of Convergence in Infinite Dimensional Holomorphy
    Domains of convergence in infinite dimensional holomorphy Von der Fakult¨at f¨urMathematik und Naturwissenschaften der Carl von Ossietzky Universit¨atOldenburg zur Erlangung des Grades und Titels eines Doktors der Naturwissenschaften (Dr. rer. nat.) angenommene Dissertation von Christopher Prengel geboren am 28.10.1975 in Hannover Gutachter: apl. Prof. Dr. A. Defant Zweitgutachter: Prof. Dr. P. Pflug Tag der Disputation: 26.10.2005 Abstract For certain subsets F(R) of holomorphic functions on a Reinhardt domain m R in a Banach sequence space X, mainly for H∞(R), P (X) and P(X), we give precise descriptions of the domain of convergence dom F(R), i.e. the set of all points of R where the monomial expansion X ∂αf(0) zα α! (N) α∈N0 of every function f ∈ F(R) is (unconditionally) convergent. The results are obtained by an interplay of complex analysis and local Banach space theory, improving work of Ryan and Lempert about monomial expan- sions of holomorphic functions on `1 and using methods from the theory of multidimensional Bohr radii. The problem of conditional convergence is solved for polynomials. It is closely connected with bases for full and symmetric tensor products and convergence preserving permutations. Zusammenfassung F¨urgewisse Teilmengen F(R) holomorpher Funktionen auf einem Reinhardt- m gebiet R in einem Banachfolgenraum X, vor allem f¨ur H∞(R), P (X) und P(X), geben wir eine genaue Beschreibung des Konvergenzbereichs dom F(R), d.h. der Menge aller Punkte aus R, in denen die Monomial- entwicklung X ∂αf(0) zα α! (N) α∈N0 einer jeden Funktion f ∈ F(R) (unbedingt) konvergiert.
    [Show full text]
  • An Introduction to Functional Analysis
    Appalachian State University Department of Mathematics Ralph Chikhany Introduction to Functional Analysis c 2015 A Directed Research Paper in Partial Fulfillment of the Requirements for the Degree of Master of Arts May 2015 Approved: Dr William J Cook Member Dr William C Bauldry Member Contents Introduction 1 1 Metric Spaces 3 1.1 Definition and Examples . .3 1.2 Underlying Topology . .6 1.3 Convergence and Continuity . .9 1.4 Completeness and Completion . 11 1.5 Compactness . 15 2 Banach Spaces 18 2.1 Normed Vector Spaces . 18 2.1.1 Vector Spaces . 18 2.1.2 Normed Spaces . 19 2.1.3 Bounded Linear Operators . 22 2.2 Banach Spaces . 24 2.2.1 Introduction to Differential Equations in Banach Spaces . 25 2.3 Linear Functionals and Duality . 28 3 Hilbert Spaces 31 3.1 Inner Product Spaces and Orthogonality . 31 3.2 Hilbert Spaces . 35 3.3 Orthonormal Sequences and Bases . 38 3.4 Representation Theorems . 41 3.5 Special Operators . 44 3.6 Weak-Weak∗ Topologies and Convergence . 48 Bibliography 51 Introduction Functional Analysis was founded at the beginning of the 20th century to provide a general abstract domain for some problems, namely from physics, where the issue was finding functions verifying certain properties. With this incentive in mind, functional analysis can be perceived as the study of function spaces. To place this field in relation to differential and integral calculus, we can classify the study of functions in three levels, from the most elementary to the abstract: The first level consists of studying properties of individual functions, namely their domain, extremas, concavity, asymptotes..
    [Show full text]
  • Explorations in Harmonic Analysis with Applications to Complex Function Theory and the Heisenberg Group
    Explorations in Harmonic Analysis with Applications to Complex Function Theory and the Heisenberg Group by Steven G. Krantz with the assistance of Lina Lee August 16, 2007 To the memory of Alberto P. Cald´eron. Table of Contents Preface v 1 Ontology and History of Real Analysis 1 1.1 DeepBackgroundinReal AnalyticFunctions. 2 1.2 The Idea of Fourier Expansions . 4 1.3 Differences of the Real Analytic Theory and the Fourier Theory 6 1.4 ModernDevelopments . .. ... .. .. .. .. ... .. .. 7 1.5 WaveletsandBeyond.. .. ... .. .. .. .. ... .. .. 8 1.6 HistoryandGenesisofFourierSeries . 9 1.6.1 DerivationoftheHeatEquation. 12 2 Advanced Ideas: The Hilbert Transform 17 2.1 TheNotionoftheHilbertTransform . 18 2.1.1 TheGutsoftheHilbertTransform . 20 2.1.2 The Laplace Equation and Singular integrals on RN . 22 2.1.3 Boundedness of the Hilbert Transform . 24 2.1.4 Lp Boundedness of the Hilbert Transform . 32 2.2 TheModifiedHilbertTransform . 33 3 Essentials of the Fourier Transform 41 3.1 Essential Properties of the Fourier Transform . 42 3.2 InvarianceandSymmetryProperties . 44 3.3 Convolution and Fourier Inversion . 50 3.4 QuadraticIntegralsandPlancherel . 58 3.5 SobolevSpaceBasics . 61 i ii 4 Fourier Multipliers 71 4.1 BasicsofFractionalIntegrals. 72 4.2 The Concept of Singular Integrals . 74 4.3 Prolegomena to Pseudodifferential Operators . 77 5 Fractional and Singular Integrals 79 5.1 Fractional and Singular Integrals Together . 80 5.2 FractionalIntegrals . 82 5.3 Background to Singular Integrals . 85 5.4 MoreonIntegralOperators . 93 6 Several Complex Variables 95 6.1 Plurisubharmonic Functions . 96 6.2 Fundamentals of Holomorphic Functions . 103 6.3 NotionsofConvexity . .106 6.3.1 The Analytic Definition of Convexity .
    [Show full text]
  • Lecture Notes on Several Complex Variables
    Lecture notes on several complex variables Harold P. Boas Draft of December 3, 2013 Contents 1 Introduction1 1.1 Power series.....................................2 1.2 Integral representations...............................3 1.3 Partial differential equations............................3 1.4 Geometry......................................4 2 Power series6 2.1 Domain of convergence...............................6 2.2 Characterization of domains of convergence....................7 2.3 Elementary properties of holomorphic functions.................. 11 2.4 The Hartogs phenomenon.............................. 12 2.5 Natural boundaries................................. 14 2.6 Summary: domains of convergence........................ 22 2.7 Separate holomorphicity implies joint holomorphicity............... 23 3 Holomorphic mappings 27 3.1 Fatou–Bieberbach domains............................. 27 3.1.1 Example.................................. 28 3.1.2 Theorem.................................. 30 3.2 Inequivalence of the ball and the bidisc...................... 31 3.3 Injectivity and the Jacobian............................. 32 3.4 The Jacobian conjecture.............................. 35 4 Convexity 37 4.1 Real convexity.................................... 37 4.2 Convexity with respect to a class of functions................... 38 4.2.1 Polynomial convexity............................ 39 4.2.2 Linear and rational convexity........................ 44 4.2.3 Holomorphic convexity........................... 47 4.2.4 Pseudoconvexity.............................
    [Show full text]