Neste trabalho aplicamos técnicas de combinatória infinitária e forcing na teoria dos espaços de Banach, investigando propriedades dos espaços de Banach da forma C(K), formado pelas funções reais contínuas sobre K com a norma do supremo, com poucos operadores, no sentido de que todo operador em C(K) é da forma gI+S, onde I é o operador identidade, g pertence a C(K) e S é fracamente compacto. Enfatizamos as construções onde K é conexo, o que implica que C(K) é indecomponível. Assumindo Axioma Diamante, um axioma combinatório mais forte que a Hipótese do Contínuo, construímos um espaço de Banach C(K) tal que C(L) tem poucos operadores, para todo L subespaço fechado de K. Sob a Hipótese do Contínuo construímos um espaço C(K) indecomponível com poucos operadores tal que K contém $\\beta N$ homeomorficamente. Em ZFC construímos um espaço C(K) com poucos operadores em um sentido estritamente mais fraco. Também mostramos a existência de pelo menos contínuo espaços de Banach C(K) indecomponíveis dois a dois essencialmente incomparáveis. Usando forcing provamos que existe consistentemente um espaço de Banach C(K) de densidade menor que contínuo com poucos operadores e um C(K) indecomponível de densidade menor que contínuo. / In this work we apply techniques of infinitary combinatorics and forcing in Banach spaces theory, investigating the compact topological spaces K such that the Banach space C(K), consisting of the continuous real-valued functions on K with the supremum norm, has few operators, in the sense that all operators on C(K) have the form gI+S, where I is the identity operator, g\\ belongs to C(K) and S is weakly compact. We emphasize the constructions where K is connected, which implies that C(K) is indecomposable. Assuming Diamond Axiom, a combinatoric axiom stronger than the continuum hypothesis, we construct a Banach space C(K) where C(L) has few operators, for every L closed subspace of K. Under continuum hypothesis we construct an indecomposable C(K) with few operators such that K contains $\\beta \\mathbb$ homeomorphically. In ZFC we construct a space C(K) with few operators in a strictly weaker sense. We also show the existence of at least continuum pairwise essentially incomparable indecomposable Banach spaces C(K). Using forcing, we prove that there exists consistently a Banach space C(K) of density smaller than continuum having few operators and an indecomposable C(K) of density smaller than continuum.

Para um espaço localmente compacto K e um espaço de Banach X, seja C_0(K,X) o espaço das funções continuas que se anulam no infinito munido da norma do supremo. Nesta tese se provam resultados relacionados com a geometria destes espaços. / For a locally compact Hausdorff space K and a Banach spaces X, let C_0(K,X) be the Banach space of continuous functions which vanish at infinity endowed with the supremum norm. We prove some results about geometry of these spaces.

Para um espaço localmente compacto K e um espaço de Banach X, seja C_0(K,X) o espaço das funções continuas que se anulam no infinito munido da norma do supremo. Nesta tese se provam resultados relacionados com a geometria destes espaços. / For a locally compact Hausdorff space K and a Banach spaces X, let C_0(K,X) be the Banach space of continuous functions which vanish at infinity endowed with the supremum norm. We prove some results about geometry of these spaces.

This thesis presents the strong and weak forms of the Banach-Tarski paradox based on the Hausdorff paradox. It provides modernized proofs of the paradoxes and necessary properties of equidecomposable and paradoxical sets. The historical significance of the paradox for measure theory is covered, along with its incorrect attribution to Banach and Tarski. Finally, the necessity of the axiom of choice is discussed and contrasted with other axiomatic and topological assumptions that enable the paradoxes. / Den här uppsatsen presenterar den starka och svaga formen av Banach-Tarskis paradox baserade på Hausdorffs paradox. Den tillhandahåller moderniserade bevis av paradoxerna och nödvändiga egenskaper av likuppdelningsbara och paradoxala mängder. Den historiska betydelsen av paradoxen på måtteori tas upp samt dess felaktiga tillskrivning till Banach och Tarski. Till sist diskuteras behovet av urvalsaxiomet som ställs i kontrast mot andra axiomatiska och topologiska antaganden som möjliggör paradoxerna.

<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>

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> 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. 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.

In this paper we give a characterization theorem for the reciprocal Dunford-Pettis property as defined by Grothendieck. The relationship of this property to Pelczynski's property V is examined. In particular it is shown that every Banach space with property V has the reciprocal Dunford-Pettis property and an example is given to show that the converse fails to hold. Moreover the characterizations of property V and the reciprocal Dunford-Pettis property lead to the definitions of property V* and property RDP* respectively. Me compare and contrast results for the reciprocal Dunford-Pettis property and property RDP* with those for properties V and V*. In the final chapter we use a result of Brooks to obtain a characterization for the Radon-Nikodým property.

Banach's contraction principle is probably one of the most important theorems in fixed point theory. It has been used to develop much of the rest of fixed point theory. Another key result in the field is a theorem due to Browder, Göhde, and Kirk involving Hilbert spaces and nonexpansive mappings. Several applications of Banach's contraction principle are made. Some of these applications involve obtaining new metrics on a space, forcing a continuous map to have a fixed point, and using conditions on the boundary of a closed ball in a Banach space to obtain a fixed point. Finally, a development of the theorem due to Browder et al. is given with Hilbert spaces replaced by uniformly convex Banach spaces.

No description available.

Ingeniero Civil Matemático / En la presente memoria se estudia la propiedad de regularidad asintótica para una variante de la iteración de \textit{Krasnoselskii-Mann} en un espacio de Banach general. Este problema está enmarcado en la teoría métrica de puntos fijos de operadores no expansivos, pues resulta ser que, bajo ciertas hipótesis, la sucesión de iterados de Krasnoselskii-Mann converge a un punto dijo de cierto operador $T$.\\ La regularidad asintótica de la iteración de Krasnoselskii-Mann ha sido ampliamente estudiada por muchos autores, ya que sirve para aproximar puntos fijos de un operador no expansivo $T:C\to C$ definido sobre un conjunto no vacío, convexo, cerrado y acotado en un espacio de Banach. Se ha establecido la regularidad asintótica de los iterados de Krasnosleskii-Mann en un espacio de Banach bajo condiciones simples, y además se conoce la tasa de regularidad asintótica en un espacio de Banach general. En esta memoria se prueba la regularidad asintótica de la iteración $$x_{k+1}=(1-\alpha_{k+1})x_k+\alpha_{k+1}(Tx_k+e_{k+1}),$$ donde $x_0\in C$, $(\alpha_k)_{k\in\N}\subseteq[0,1]$, $e_n\to 0$, $\sum\alpha_k(1-\alpha_k)=\infty$ y $\sum\alpha_k\|e_k\|<\infty$. Además, se establece la tasa de convergencia de $\|x_n-Tx_n\|$ cuando los coeficientes $(\alpha_k)_{k\in\N}$ están lo suficientemente alejados de $0$ y $1$. Por último, se aplican los resultados obtenidos en esta tesis para estudiar la regularidad asintótica de otros procesos iterativos y para estudiar cotas de la solución de una ecuación de evolución no lineal. / Este trabajo ha sido parcialmente financiado por el proyecto Fondecyt 1130564