Thesis (Ph. D.)--University of Wisconsin--Madison, 1951. / Typescript. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references (leaves 46-47).

Let X be a Hausdorff locally convex space, U be a base for closed absolutely convex O-neighborhoods in X , K C X be nonempty. For each U є U , we denote by P[subscript u] the gauge of U. Then T : K ↦ K is said to be nonexpansive w.r.t. U if and only if for each U є U, P[subscript u](T(x) - T(y)) ≤ P[subscript u](x - y) for all x, y є K; T: K ↦ K is said to be strictly contractive w.r.t. U if and. only if for each U є U, there is a constant λ[subscript u] with 0 ≤ λ[subscript u] < 1 such that P[subscript u](T(x) - T(y)) ≤λ[subscript u]P[subscript u](x - y) for all x, y є K . The concept of nonexpansive (respectively strictly contractive) mappings is originally defined on a metric space. The above definitions are generalizations if the topology on X is induced by a translation invariant metric, and in particular if X is a normed space. An analogue of the Banach contraction mapping principle is proved and some examples together with an implicit function theorem are shown as applications. Moreover several fixed point theorems for various kinds of nonexpansive mappings are obtained. The convergence of nets of nonexpansive mappings and that of fixed points are also studied. Finally on sets with 'complete normal structure', a common fixed point theorem is obtained for an arbitrary family of 'commuting' nonexpansive mappings while on sets with 'normal structure', a common fixed point, theorem for an arbitrary family of (not necessarily commuting) 'weakly periodic' nonexpansive mappings is obtained. / Science, Faculty of / Mathematics, Department of / Graduate

by Ngai-ching Wong. / Thesis (M.Ph.)--Chinese University of Hong Kong, 1987. / Bibliography: leaves 197-201.

Metric topologies can be viewed as one-dimensional measures. This dissertation is a topological study of two-dimensional measures. Attention is focused on locally convex vector topologies on infinite dimensional real spaces. A nabla (referred to in the literature as a 2-norm) is the analogue of a norm which assigns areas to the parallelograms. Nablas are defined for the classical normed spaces and techniques are developed for defining nablas on arbitrary spaces. The work here brings out a strong connection with tensor and wedge products. Aside from the normable theory, it is shown that nabla topologies need not be metrizable or Mackey. A class of concretely given non-Mackey nablas on the ℓp and Lp spaces is introduced and extensively analyzed. Among other results it is found that the topological dual of ℓ₁ with respect to these nabla topologies is C₀, one of the spaces infamous for having no normed predual. Also, a connection is made with the theory of two-norm convergence (not to be confused with 2-norms). In addition to the hard analysis on the classical spaces, a duality framework from which to study the softer aspects is introduced. This theory is developed in analogy with polar duality. The ideas corresponding to barrelledness, quasi-barrelledness, equicontinuity and so on are developed. This dissertation concludes with a discussion of angles in arbitrary normed spaces and a list of open questions.

Song-Jian Han. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1994. / Includes bibliographical references (leaves 72-73). / Acknowledgement --- p.i / Abstract --- p.ii / Introduction --- p.iii / Chapter 1 --- Topological Vector Spaces and Elemantary Duality Theory --- p.1 / Chapter 1.1 --- Locally Convex Spaces --- p.2 / Chapter 1.2 --- Bornological Spaces and Bornological Vector Spaces --- p.4 / Chapter 1.3 --- Elementary Properties of Dual Spaces --- p.6 / Chapter 1.4 --- Topological Injections and Surjections Bornological Injections and Surjections --- p.10 / Chapter 2 --- Locally Convex Riesz Spaces --- p.15 / Chapter 2.1 --- Ordered Vector Spaces --- p.15 / Chapter 2.2 --- Riesz Space --- p.18 / Chapter 2.3 --- Locally Convex Riesz Spaces --- p.20 / Chapter 3 --- Half-Full Injections and Half-Decomposable Surjections Half- Full Bornological Injections and Half-Decomposable Bornologi- cal Surjections --- p.24 / Chapter 4 --- Operator Modules between Locally Convex Riesz Spaces --- p.35 / Chapter 4.1 --- Preliminaries --- p.35 / Chapter 4.2 --- Operator Modules and Ideal Cones --- p.37 / Chapter 4.3 --- The Half-Full Injective Hull and the Half-Decomposable Bornolog- ical Surjective Hull of Operator Modules Between Locally Convex Riesz Spaces --- p.41 / Chapter 4.4 --- Extensions of Operator Modules and Ideal Cones --- p.57 / References --- p.72

Orientador: Mario Carvalho de Matos / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-10T04:31:29Z (GMT). No. of bitstreams: 1 Favaro_ViniciusVieira_D.pdf: 758368 bytes, checksum: 1618ae17ed57109a9f307089c59fdbb6 (MD5) Previous issue date: 2007 / Resumo: Neste trabalho introduzimos os espaços de funções (s , m (r , q))-somantes de um dado tipo e uma dada ordem, definidas em E, e os espaços de funções (s; (r; q))-quase-nucleares de um dado tipo e uma dada ordem, definidas em E, e provamos que a transformada de Fourier-Borel identica o dual do espaço de funções (s ; (r' , q'))-quase-nucleares de um dado tipo e uma dada ordem, definidas em E, com o espaço de funções (s';m (r' , q'))-somantes de um correspondente tipo e uma correspondente ordem, definidas em E'. Provamos também teoremas de divisão para funções (s ; m (r , q))-somantes de um dado tipo e uma dada ordem e teoremas de divisão envolvendo a transformada de Fourier-Borel. Como consequencia, provamos resultados de existência e aproximação de soluções de equações de convulção nos espaços de funções (s; (r , q))-quase-nucleares de um dado tipo e uma dada ordem / Abstract: In this work we introduce the spaces of (s; m (r , q))-summing functions of a given type and order defined in E, and the spaces of (s; (r, q))-quasi-nuclear functions of a given type and order defined in E, and we prove that the Fourier-Borel transform identify the dual of the space of (s; (r; q))-quasi-nuclear functions of a given type and order defined in E, with the space of (s' ; m (r'; q'))-summing functions of a corresponding type and order defined in E'. We also prove division theorems for (s; m (r; q))-summing functions of a given type and order and division theorems involving the Fourier-Borel transform. As a consequence we prove the existence and approximation results for convolution equations on the spaces of (s; (r; q))-quasi-nuclear functions of a given type and order / Doutorado / Analise Funcional / Doutor em Matemática

Orientador: Jorge Tulio Mujica Ascui / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-07T16:44:31Z (GMT). No. of bitstreams: 1 Takatsuka_Paula_D.pdf: 3540981 bytes, checksum: 643e40ac81900cf042cfe1cfb3737b0d (MD5) Previous issue date: 2006 / Resumo: Este trabalho estende teoremas clássicos da teoria de funções holomorfas de uma variável complexa para espaços localmente convexos de dimensão infinita. Serão dadas várias caracterizações de famílias normais, n¿ao apenas com relação à topologia compacto-aberta, mas também para outras topologias naturais no espaço de aplicações holomorfas. Teoremas de tipo Montel e de tipo Schottky, bem como outros resultados correlatos, ser¿ao estabelecidos em dimensão infinita para as diferentes topologias. Teoremas de limita¸c¿ao universal sobre famílias de funções holomorfas que omitem dois valores distintos ser¿ao formulados para espaços de Banach / Abstract: The present work extends some classical theorems from the theory of holomorphic functions of one complex variable to infinite dimensional locally convex spaces. Several characterizations of normal families are given, not only for the compact-open topology, but also for other natural topologies on spaces of holomorphic mappings. Montel-type and Schottky-type theorems and various related results are established in infinite dimension for these different topologies. Universal boundedness theorems concerning families of holomorphic functions which omit two distinct values are formulated for Banach spaces / Doutorado / Mestre em Matemática

[EN] The Ph.D. Thesis "Atomic decompositions and frames in Fréchet spaces and their duals" presented here treats different areas of functional analysis with applications. Schauder frames are used to represent an arbitrary element x of a function space E as a series expansion involving a fixed countable set {xj} of elements in that space such that the coefficients of the expansion of x depend in a linear and continuous way on x. Unlike Schauder bases, the expression of an element x in terms of the sequence {xj}, i.e. the reconstruction formula for x, is not necessarily unique. Atomic decompositions or Schauder frames are a less restrictive structure than bases, because a complemented subspace of a Banach space with basis has always a natural Schauder frame, that is obtained from the basis of the superspace. Even when the complemented subspace has a basis, there is not a systematic way to find it. Atomic decompositions appeared in applications to signal processing and sampling theory among other areas. Very recently, Pilipovic and Stoeva [55] studied series expansions in (countable) projective or inductive limits of Banach spaces. In this thesis we begin a systematic study of Schauder frames in locally convex spaces, but our main interest lies in Fréchet spaces and their duals. The main difference with respect to the concept considered in [55] is that our approach does not depend on a fixed representation of the Fréchet space as a projective limit of Banach spaces. The text is divided into two chapters and appendix that gives the notation, definitions and the basic results we will use throughout the thesis. The first one focuses on the relation between the properties of an existing Schauder frame in a Fréchet space E and the structure of the space. In the second chapter frames and Bessel sequences in Fréchet spaces and their duals are defined and studied. In what follows, we give a brief description of the different chapters: In Chapter 1, we study Schauder frames in Fréchet spaces and their duals, as well as perturbation results. We define shrinking and boundedly complete Schauder xviiframes on a locally convex space, study the duality of these two concepts and their relation with the reflexivity of the space. We characterize when an unconditional Schauder frame is shrinking or boundedly complete in terms of properties of the space. Several examples of concrete Schauder frames in function spaces are also presented. Most of the results included in this chapter are published by Bonet, Fernández, Galbis and Ribera in [13]. The second chapter of the thesis is devoted to study ¿-Bessel sequences, ¿-frames and frames with respect to ¿ in the dual of a Hausdorff locally convex space E, in particular for Fréchet spaces and complete (LB)-spaces E, with ¿ a sequence space. We investigate the relation of these concepts with representing systems in the sense of Kadets and Korobeinik [34] and with the Schauder frames, that were investigated in Chapter 1. The abstract results presented here, when applied to concrete spaces of analytic functions, give many examples and consequences about sampling sets and Dirichlet series expansions. We present several abstract results about ¿-frames in complete (LB)-spaces. Finally, many applications, results and examples concerning sufficient sets for weighted Fréchet spaces of holomorphic functions and weakly sufficient sets for weighted (LB)-spaces of holomorphic functions are collected. Most of the results are submitted for publication in a preprint of Bonet, Fernández, Galbis and Ribera in [12]. / [ES] La presente memoria "Descomposiciones atómicas y frames en espacios de Fréchet y sus duales" trata diferentes áreas del análisis funcional con aplicaciones. Los frames de Schauder se utilizan para representar un elemento arbitrario x de un espacio de funciones E mediante una serie a partir de un conjunto numerable fijado {xj} de elementos de este espacio de manera que los coeficientes de la reconstrucción de x dependen de forma lineal y continua de x. A diferencia de las bases de Schauder, la expresión de un elemento x en términos de la sucesión {xj}, i.e. la fórmula de reconstrucción para x, no es necesariamente única. Las descomposiciones atómicas o los frames de Schauder son un estructura menos restrictiva que las bases, porque un subespacio complementado de un espacio de Banach con base tiene siempre un frame de Schauder natural, que se obtiene a partir de una base del superespacio. Incluso cuando el subespacio complementado tiene una base, no hay una forma sistemática de encontrarla. Las descomposiciones atómicas aparecen en aplicaciones al procesamiento de señales y la teoría de muestreo, entre otras áreas. Recientemente, Pilipovic y Stoeva [55] han estudiado el desarrollo en serie en límites inductivos y proyectivos (numerables) de espacios de Banach. En esta tesis empezamos un estudio sistemático de los frames de Schauder en espacios localmente convexos aunque nuestro interés principal son los espacios de Fréchet y sus duales. La diferencia principal respecto del concepto considerado en [55] es que nuestra aproximación no depende de una representación fijada del espacio de Fréchet como límite proyectivo de espacios de Banach. El texto queda dividido en dos partes y un apéndice que incluye la notación, las definiciones y los resultados básicos que usaremos a lo largo de la tesis. La primera parte se centra en la relación entre las propiedades de un frame de Schauder en un espacio de Fréchet E y la estructura del espacio. En el segundo capítulo se definen y estudian los frames y las sucesiones de Bessel en espacios de Fréchet y sus duales. A continuación, presentamos una breve descripción de los capítulos: En el Capítulo 1, estudiamos los frames de Schauder en los espacios de Fréchet y sus duales así como los resultados de perturbación. Definimos los frames de Schauder contractivos y acotadamente completos en espacios localmente convexos, estudiamos la dualidad de estos dos conceptos y su relación con la reflexividad del espacio. Caracterizamos cuándo un frame de Schauder incondicional es contractivo o acotadamente completo en términos de las propiedades del espacio. También se presentan varios ejemplos de frames de Schauder en espacios de funciones concretos. La mayoría de los resultados incluidos en este capítulo están publicados por Bonet, Fernández, Galbis y Ribera en [13]. El segundo capítulo de la tesis está centrado en el estudio de las sucesiones de ¿-Bessel, ¿-frames y frames respecto de ¿ en el dual de un espacio localmente convexo de Hausdorff E, en particular, para espacios de Fréchet y espacios (LB) completos E, con ¿ un espacio de sucesiones. Investigamos la relación de estos dos conceptos con los sistemas representantes en el sentido de Kadets y Korobeinik [34] y con los frames de Schauder, considerados en el Capítulo 1. Los resultados abstractos presentados aquí, cuando los aplicamos a espacios de funciones analíticas concretos, nos dan muchos ejemplos y consecuencias sobre los conjuntos de muestreo y los desarrollos en serie de Dirichlet. Presentamos varios resultados abstractos sobre ¿-frames en espacios (LB) completos. Finalmente, recogemos muchas aplicaciones, resultados y ejemplos alrededor de los conjuntos suficientes para espacios de Fréchet de funciones holomorfas y conjuntos débilmente suficientes para espacios pesados (LB) de funciones holomorfas. La mayoría de los resultados incluidos en este capítulo están enviados para publicar e / [CA] La tesi "Descomposicions atòmiques i frames en espais de Fréchet i els seus duals" presentada ací tracta diferents àrees de l'anàlisi funcional amb aplicacions. Els frames de Schauder s'utilitzen per tal de representar un element arbitrari x d'un espai de funcions E com una reconstrucció en sèrie a partir d'un conjunt numerable fixat {xj} d'elements en aquest espai tal que els coeficients de la reconstrucció de x depenen de forma lineal i continua de x. A diferència de les bases de Schauder, l'expressió d'un element x en termes d'una successió {xj}, i.e. la fórmula de reconstrucció per a x, no és necessàriament única. Les descomposicions atòmiques o els frames de Schauder són una estructura menys restrictiva que les bases, donat que un subespai complementat d'un espai de Banach amb base sempre té un frame de Schauder natural, el qual és obtingut a partir d'una base del superespai. Inclòs quan el subespai complementat disposa de una base, no hi ha una forma sistemàtica per tal de trobar-la. Les descomposicions atòmiques apareixen en aplicacions a processat de senyals i teoría de mostreig entre altres àrees. Recentment, Pilipovic i Stoeva [55] han estudiat els desenvolupaments en sèrie en límits inductius o projectius (numerables) en espais de Banach. En aquesta tesi comencem un estudi sistemàtic dels frames de Schauder en espais localment convexos, tot i que el nostre interés està en els espais de Fréchet i els seus duals. La diferència més important amb el concepte estudiat en [55] és que el nostre estudi no depén de una representació fixada del espai de Fréchet com a límit projectiu de espais de Banach. El text està dividit en dos capítols i un apèndix que ens aporta la notació, definicions i els resultats bàsics que utilitzarem al llarg de la tesi. El primer dels capítols està centrat en la relació entre les propietats de un frame de Schauder en un espai de Fréchet E i la estructura del espai. En el segon capítol es defineixen i estudien els frames i les successions de Bessel en espais de Fréchet i els seus duals. En el que segueix, donem una breu descripció dels diferents capítols: En el Capítol 1, estudiem els frames de Schauder en els espais de Fréchet i els seus duals, així com els resultats de pertorbació. Definim els frames de Schauder contractius i fitadament complets en espais localment convexos, estudiem la dualitat d'aquests dos conceptes i la seua relació amb la reflexivitat del espai. Caracteritzem, en quines situacions, un frame de Schauder incondicional és contractiu o fitadament complet en termes de les propietats del espai. També presentem alguns exemples de frames de Schauder concrets en espais de funcions. La majoria dels resultats inclosos en aquest capítol estan publicats per Bonet, Fernández, Galbis i Ribera en [13]. El segon capítol de la tesi està centrat en el estudi de les successions ¿-Bessel, ¿-frames i frames respecte de ¿ en el dual d'un espai localment convex de Hausdorff E, en particular, per a espais de Fréchet i espais (LB) complets E, amb ¿ un espai de successions. Investiguem la relació d'aquests dos conceptes amb sistemes representants en el sentit de Kadets i Korobeinik [34] i amb els frames de Schauder, que han sigut investigats en el Capítol 1. Els resultats abstractes presentats ací, quan els apliquem a espais de funcions analítiques concrets, ens donen molts exemples i conseqüències sobre els conjunts de mostreig i els desenvolupaments en sèrie de Dirichlet. Presentem diversos resultats abstractes sobre ¿-frames en espais (LB) complets. Finalment, recollim moltes aplicacions, resultats i exemples al voltant dels conjunts suficients per a espais de Fréchet de funcions holomorfes i conjunts dèbilment suficients per a espais pesats (LB) de funcions holomorfes. La majoria dels resultats inclosos en aquest capítol estan sotmesos a publicació per Bonet, Fernández, Galbis i Ribera en [12]. / Ribera Puchades, JM. (2015). Atomic decompositions and frames in Fréchet spaces and their duals [Tesis doctoral]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/49987

Submitted by ANA KARLA PEREIRA RODRIGUES (anakarla_@hotmail.com) on 2017-08-17T14:00:23Z No. of bitstreams: 1 arquivototal.pdf: 1661057 bytes, checksum: 913a7f671e2e028b60d14a02274f932a (MD5) / Made available in DSpace on 2017-08-17T14:00:23Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 1661057 bytes, checksum: 913a7f671e2e028b60d14a02274f932a (MD5) Previous issue date: 2015-02-27 / In this work we investigate the concept of topological vector spaces and their properties. In the rst chapter we present two sections of basic results and in the other sections we present a more general study of such spaces. In the second chapter we restrict ourselves to the real scalar eld and we study, in the context of locally convex spaces, the Hahn-Banach and Banach-Alaoglu theorems. We also build the weak, weak-star, of bounded convergence and of pointwise convergence topologies. Finally we investigate the Theorem of Banach-Steinhauss, the Open Mapping Theorem and the Closed Graph Theorem. / Neste trabalho, estudamos o conceito de espa cos vetoriais topol ogicos e suas propriedades. No primeiro cap tulo, apresentamos duas se c~oes de resultados b asicos e, nas demais se c~oes, apresentamos um estudo sobre tais espa cos de forma mais ampla. No segundo cap tulo, restringimo-nos ao corpo dos reais e fazemos um estudo sobre os espa cos localmente convexos, o Teorema de Hahn-Banach, o Teorema de Banach- Alaoglu, constru mos as topologias fraca, fraca-estrela, da converg^encia limitada e da converg^encia pontual. Por ultimo, estudamos o Teorema da Limita c~ao Uniforme, o Teorema do Gr a co Fechado e o da Aplica c~ao Aberta no contexto mais geral dos espa cos de Fr echet.