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This paper develops an integral for Lebesgue measurable functions mapping from the interval [0, 1] into a Banach space.

Let $mathcal{B}$ be a separable Banach space. In this thesis, it is shown that the Chung's strong law of large numbers holds for a sequence of independent $mathcal{B}$-valued random elements and an array of rowwise independent $mathcal{B}$-valued random elements under some weaker assumptions by using more generalized functions $phi_{n}$'s.

"A natural process in examining properties of Banach spaces is to see if a Banach space can be decomposed into simpler Banach spaces; in other words, to see if a Banach space has complemented subspaces. This thesis concentrates on three main aspects of this problem: norm of projections of a Banach space onto its finite dimensional subspaces; a class of Banach spaces, each of which has a large number of infinite dimensional complemented subspaces; and methods of finding Banach spaces which have uncomplemented subspaces, where the subspaces and the quotient spaces are chosen as well-known classical sequence spaces (finding non-trivial twisted sums)." --Abstract. / Master of Mathematical Sciences

The structure and geometry of subspaces of a given Banach space is among the most fundamental questions in Functional Analysis. In 1961, Kadec and Pelczyński pioneered a field of study by analyzing the structures of subspaces and basic sequences in L_p[0,1] under a naturally occurring restriction of p, 2 < p <\infty. They proved that any infinite-dimensional subspace X\subset L_p[0,1] for 2<p<\infty must either be isomorphic to l_2 and complemented in L_p or must contain a complemented subspace which is isomorphic to l_p. Many works since have studied the relationships between the sides of this dichotomy, chiefly by weakening hypotheses on side of the equation to gain stronger assumptions on the other. In this way, Johnson and Odell were able to show in 1974 that if X contains no further subspace which is isomorphic to l_2, then it must embed into l_p. Kalton and Werner further strengthened this result in 1993 by showing that such an embedding must be almost isometric. We start by analyzing the tools and definitions originally introduced in 1961 and define a natural extension to these methods. By analyzing this extension, we provide a constructive and streamlined reproving of Kalton and Werner's theorem: Let X be an infinite dimensional subspace of L_p[0,1] for 2<p<\infty. Then, either X contains a subspace which is isomorphic to l_2, or for every \varepsilon>\ 0, X embeds into l_p with constant 1 + \varepsilon.

Orientador: Jorge Tulio Mujica Ascui / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-08T22:02:35Z (GMT). No. of bitstreams: 1 Kuo_PoLing_D.pdf: 378648 bytes, checksum: 3724407371cc36985fd2257b4bc5fe8c (MD5) Previous issue date: 2007 / Resumo: Este trabalho está dedicado ao estudo dos operadores de Nicodemi, introduzidos em [7] a partir de uma idéia em [12]. Os operadores de Nicodemi levam aplicações multilineares (resp. polinômios homogêneos) de um espaço de Banach E em aplicações multilineares (resp. polinômios homogêneos) em um espaço de Banach F. O nosso primeiro objetivo é encontrar condições para que os operadores de Nicodemi preservem certos tipos de aplicações multilineares (resp. polinômios homogêneos). Em particular estudamos a preservação de aplicações multilineares simétricas, de tipo finito, nucleares, compactas ou fracamente compactas. O segundo objetivo é encontrar condições para que, se os espaços duais E¿ e F¿ são isomorfos, os espaços de aplicações multilineares (resp. polinômios homogêneos) em E e F sejam isomorfos também. Estudamos também o problema correspondente para os espaços de aplicações multilineares (resp. polinômios homogêneos) de um determinado tipo, como por exemplo, de tipo finito, nuclear, compacto ou fracamente compacto / Abstract: This work is devoted to studying the Nicodemi operators, introduced in [7], following an idea in [12]. The Nicodemi operators map multilinear mappings (resp. homogeneous polynomials) on a Banach spaces E into multilinear mappings (resp. homogeneous polynomials) on a Banach spaces F. Our first objective is to find conditions under which the Nicodemi operators preserve certain types of multilinear mappings (resp. homogeneous polynomials). In particular we examine the preservation of the multilinear mappings that are symmetric, of finite type, nuclear, compact or weakly compact. Our second objective is tofind conditions under which, whenever the dual spaces E¿ and F¿ are isomorphic, the spaces of multilinear mappings (resp. homogeneous polynomials) on E and F are isomorphic as well. We also examine the corresponding problem for the spaces of multilinear mappings (resp. homogeneous polynomials) of a certain type, for instance of finite, nuclear, compact or weakly compact type / Doutorado / Analise Funcional / Doutor em Matemática

The thesis consists of three papers of the author. In the first paper, it is shown that the sets of Fréchet subdifferentiability of Lipschitz functions on a Banach space X are Borel if and only if X is reflexive. This answers a ques- tion of L. Zajíček. In the second paper, a problem of G. Debs, G. Godefroy and J. Saint Raymond is solved. On every separable non-reflexive Banach space, equivalent strictly convex norms with the set of norm-attaining func- tionals of arbitrarily high Borel class are constructed. In the last paper, binormality, a separation property of the norm and weak topologies of a Ba- nach space, is studied. A result of P. Holický is generalized. It is shown that every Banach space which belongs to a P-class is binormal. It is also shown that the asplundness of a Banach space is equivalent to a related separation property of its dual space. 1

Let X be a Banach space and denote by SS₁(X) the set of all S₁-strictly singular operators from X to X. We prove that there is a Banach space X such that SS₁(X) is not a closed ideal. More specifically, we construct space X and operators T₁ and T₂ in SS₁(X) such that T₁+T₂ is not in SS₁(X). We show one example where the space X is reflexive and other where it is c₀-saturated. We also develop some results about S_alpha-strictly singular operators for alpha less than omega_1. / text

This thesis studies abstract stochastic differential equations on Banach spaces. The well-posedness of abstract stochastic differential equations on such spaces is a recent result of van Neerven, Veraar and Weis, based on the theory of stochastic integration of Banach space valued processes constructed by the same authors. We study existence and uniqueness for solutions of stochastic differential equations with (possibly infinite) delay in their inputs on UMD Banach spaces. Such problems are also known as functional differential equations or delay differential equations. We show that the methods of van Neerven et al. extend to such problems if the initial history of the system lies in a space of a type introduced by Hale and Kato. The results are essentially of a fixed point type, both autonomous and non-autonomous cases are discussed and an example is given. We also study some long time properties of solutions to these stochastic differential equations on general Banach spaces. We show the existence of solutions to stochastic problems with almost periodicity in a weak or distributional sense. Results are again given for both autonomous and non-autonomous cases and depend heavily on estimates for R-bounds of operator families developed by Veraar. An example is given for a second order differential operator on a domain in ℝ^d. Finally we consider the existence of invariant measures for such problems. This extends recent work of van Gaans in Hilbert spaces to Banach spaces of type 2.

The goal of this thesis is to give an exposition of the following recent result of Freeman, Lennard, Odell, Turett and Randrianantoanina. A Banach space has the Schur property if and only if every weakly compact set is contained in the closed convex hull of a weakly null sequence. This result complements an old result of Grothendieck (now called the Grothendieck Compactness Principle) stating that every norm compact subset of a Banach space is contained in the closed convex hull of a norm null sequence. We include many of the relevant definitions and preliminary results which are required in the proofs of both of these theorems.