Univerzita Karlova Abstract of the diploma thesis Analysis in Banach spaces Matěj Novotný, Praha 2013 In the thesis, connection between two certain types of equivalence on Ba- nach spaces is studied: Between Lipschitz and linear one. In general, linear equivalence of two Banach spaces implies their Lipschitz equivalence, but the converse need not be true, which is shown by some nonseparable examples. There are summarized several examples to this question in the thesis, both positive and negative ones. Moreover, it is shown that James' quasi-reflexive space and its dual space have unique Lipschitz structure. To prove this, theory of linearization of Lipschitz mappings and at the same time linear structure of the two mentioned spaces is used. 1

While the spectral theory of compact operators is known to many, knowledge regarding the relationship between eigenvalues and approximation numbers might be less known. By examining these numbers in tandem, one may develop a link between eigenvalues and l^p spaces. In this paper, we develop the background of this connection with in-depth examples.

Submitted by Carlos Alberto Do Nascimento (prof.math.edu@gmail.com) on 2018-06-04T20:15:40Z No. of bitstreams: 1 Dissertação.pdf: 2749900 bytes, checksum: d578113faba2ff3354a375ae6c4e7e1e (MD5) / Approved for entry into archive by Ana Paula Santulo Custódio de Medeiros null (asantulo@rc.unesp.br) on 2018-06-05T12:25:11Z (GMT) No. of bitstreams: 1 nascimento_ca_me_rcla.pdf: 2739624 bytes, checksum: c8831bbd095a228cff3121b4621d7091 (MD5) / Made available in DSpace on 2018-06-05T12:25:12Z (GMT). No. of bitstreams: 1 nascimento_ca_me_rcla.pdf: 2739624 bytes, checksum: c8831bbd095a228cff3121b4621d7091 (MD5) Previous issue date: 2018-05-03 / Neste trabalho, abordaremos os principais conceitos e propriedades sobre espaço de Banach e espaço de Hilbert com o objetivo de oferecer o conteúdo necessário para discutirmos algumas aplicações desses conceitos. Mostraremos a existência e unicidade de solução de Equações Diferenciais Ordinárias de Primeira Ordem, existência e unicidade de solução de certas Equações Integrais e existência e unicidade de melhor aproximação em espaços normados e de Hilbert. / In this work, we will discuss the main concepts and properties on Banach space and Hilbert space in order to offer the necessary content to discuss some applications of these concepts. We will show the existence and uniqueness of the solution of First Order Ordinary Differential Equations, existence and uniqueness of solution of certain Integral Equations and existence and uniqueness of better approximation in normed and Hilbert spaces.

This thesis consists of four research papers. In the first paper we study whether certain properties of sets (functions) are separably determined. In our results we use the "method of elementary submodels". In the second paper we generalize some results concerning Valdivia compacta (equivalently spaces with a commutative retractional skeleton) to the context of spaces with a retractional skeleton (not necessarily commutative). The third paper further studies the structure of spaces with a projectional (resp. retractional) skeleton. Under certain conditions we prove the existence of a "simultaneous projectional skeleton" and we use this result to prove other statements concerning the structure of spaces with a projectional (resp. retractional) skeleton. In the last paper we study the method of elementary submodels in a greater detail and we compare it with the "method of rich families". 1

From the beginning of the study of spaces in functional analysis, bases have been an indispensable tool for operating with vectors and functions over a concrete space. Bases can be organized by types, depending on their properties. This thesis is intended to give an overview of some bases and their relations. We study Hamel basis, Schauder basis and Orthonormal basis; we give some properties and compare them in different spaces, explaining the results. For example, an infinite dimensional Hilbert space will never have a basis which is a Schauder basis and a Hamel basis at the same time, but if this space is separable it has an orthonormal basis, which is also a Schauder basis. The project deals mainly with Banach spaces, but we also talk about the case when the space is a pre Hilbert space.

Orientador: Jorge Tulio Mujica Ascui / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-08T09:42:47Z (GMT). No. of bitstreams: 1 Miyamura_MauricioYudi_M.pdf: 634948 bytes, checksum: 09f557fd6fcd4987f744c3d9c645f602 (MD5) Previous issue date: 2007 / Resumo: Sejam E e F espaços de Banach. Os principais resultados que iremos expor serão teoremas sobre a reflexividade de L (E; F) e P (mE; F).. No capítulo 2, estudamos alguns conceitos básicos da teoria de produtos tensoriais de espaços de Banach. A importância do capítulo 2 para o trabalho seria, essencialmente, a identificação do espaço de operadores lineares contínuos L (E; F) com o dual do produto tensorial projetivo E ÄpF?. No capítulo 3, que trata de espaços de polinômios homogêneos, incluímos de noções e resultados básicos e estudamos um teorema de linearização que permitirá transferir resultados em espaços de operadores lineares para espaços de polinômios homogêneos. / Abstract: Let E and F be Banach spaces. The main results in this work are theorems concerning the reflexivity of L (E; F) and P (mE; F). In Chapter 2, we study basic concepts of the theory of tensor products of Banach spaces. The importance of Chapter 2 will be, essentially, the identification of the space of continuous linear operators L(E; F) with the dual of the projective tensor product E ÄpF?. In Chapter 3, that deals with homogeneous polynomials, we include basic definitions and results and we study a linearization theorem that will allow to transfer results from spaces of linear operators to spaces of homogeneous polynomials. / Mestrado / Matematica / Mestre em Matemática

Cette thèse est principalement consacrée à des problèmes de dynamique linéaire dans les espaces de Banach. Répondant à une question récente de Hajek et Smith, on construit notamment, dans tout espace de Banach séparable, un opérateur borné tel que ses orbites tendent vers l'infini sur une partie ni vide, ni dense. On relie également, à l'aide d'un autre résultat, le module de lissité asymptotique au comportement des opérateurs bornés. / This work is mainly devoted to some problems of linear dynamics in Banach spaces. In particular, we answer a recent question of Hajek and Smith by constructing, in any separable Banach space, a bounded operator such that its orbits tending to infinity form a set which is neither empty, nor dense. We also connect the behaviour of bounded operators with the asymptotic modulus of smoothness.

We start with the recollection of basic results about differential manifolds and Lie groups. We also recall some preliminary terminologies in Lie algebra. Then we define the Lie algebra corresponding to a Lie group. In the next section, we define a strongly continuous representation of a Lie group on a Banach space. We further define the smooth, analytic and entire vectors for a given representation. Then, we move on to develop some necessary and sufficient criteria to characterize smooth, analytic and entire vectors. We, in particular, take into account of some specific representations of Lie groups like the regular representation of R, the irreducible representations of Heisenberg groups, the irreducible representations of the group of Affine transformations and finally the representations of non-compact simple Lie groups.

Il existe de nombreuses notions de petitesse en analyse. On considère trois d'entre elles: la Haar-négligeabilité, la Gauss-négligeabilité et la sigma-porosité. On étudie à quelles conditions le cône positif d'une base de Schauder est Haar-négligeable, et ce que cela entraîne pour l'espace de Banach associé. On étudie également sous quelles conditions l'ensemble des vecteurs non-hypercycliques d'un opérateur hypercyclique est Haar-négligeable ou sigma-poreux. / There are many notions of smallness in Analysis. We will consider three of them: Haar-negligeability, Gauss-negligeability and sigma-porosity. We will study on which conditions the positive cone of a Schauder basis is Haar-null, and its consequence on the Banach space. We will also study on which conditions the set of non-hypercyclic vectors of an hypercyclic operator is Haar-null or sigma-porous.

<p>Suppose that <b>X</b> and <b>Y</b> are Banach couples and suppose that there is a bounded linear couple map Q from <b>Y</b> to <b>X</b> which has the property that Q restricted to the endpoint spaces is injective and the images of the endpointspaces of <b>Y</b> are closed in the endpoint spaces of <b>X</b>, then we say that <b>Y</b> is a subcouple of <b>X.</b> </p><p>If F is an interpolation functor we want to know how F(<b>Y</b>) is related to F(<b>X</b>). In particular we want to know for which F it holds that Q is an injection that maps F(<b>Y</b>) onto a closed subspace of F(<b>X</b>). In recent years interest has been paid to subcouples of finite codimension and in particular to subcouples of codimension one. We will in this thesis present an interpolation theory for subcouples of codimension one and then generalize it to finite codimension. Our theory will include both a larger class of couples and a larger class of interpolation functors than earlier results.</p><p>The interpolation method that will be considered is the regular real method. Our general theory will imply older results by Kalton, Ivanov and Löfström. We will use the theory to answer questions about Hardy-type inequalities that were raised by Krugljak, Maligranda and Persson in 1999 and our new theory will also answer a question concerning interpolation of Banach algebras.</p>