by Ming-Wai Cheung. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1992. / Includes bibliographical references (leaves 64-65). / Introduction --- p.1 / Chapter Chapter 0. --- Notations and Preliminaries --- p.3 / Chapter 0.0. --- Bornologies / Chapter 0.1. --- Ordered Vector Spaces / Chapter 0.2. --- Vector Lattices / Chapter 0.3. --- Substructures of vector lattices and Order Completeness / Chapter 0.4. --- Banach Lattices / Chapter 0.5. --- AL- and AM- spaces / Chapter 0.6. --- Operator Modules and Ideal Cones / Chapter Chapter 1. --- Tensor Products of Ordered Banach Spaces --- p.25 / Chapter Chapter 2. --- Positive Approximation Property --- p.30 / Chapter 2.1. --- Approximation Properties in LCS / Chapter 2.2. --- Approximation Properties in Banach Spaces / Chapter 2.3. --- Positive Approximation Properties in Locally Solid Spaces / Chapter 2.4. --- Positive A.P. in Locally Solid OBS and Banach Lattices / Chapter Chapter 3. --- Order Bounded Bornologies of Banach Lattices --- p.49 / Chapter 3.1. --- Characterizations of B-Lattices by the Order-Bounded Bornologies / Chapter 3.2. --- Order-Bounded-Compact Property of Banach Lattices / References --- p.64 / Notations --- p.67

by Wai-Chiu Cheung. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1992. / Includes bibliographical references (leaves 117-119). / Introduction --- p.1 / Chapter Chapter 1 --- Topological Riesz Spaces --- p.5 / Chapter 1.1 --- Locally convex spaces --- p.5 / Chapter 1.2 --- Ordered vector spaces and Riesz spaces --- p.10 / Chapter 1.3 --- Locally convex Riesz spaces --- p.16 / Chapter 1.4 --- Banach lattices --- p.23 / Chapter Chapter 2 --- Operator Modules and Ideal Cones --- p.32 / Chapter 2.1 --- Operator Modules and Ideal Cones on Banach lattices --- p.32 / Chapter 2.2 --- Half-injective hull and half-surjective hull of operator modules and ideal cones --- p.38 / Chapter 2.3 --- Topologies determined by operator modules and ideal cones --- p.49 / Chapter 2.4 --- Bornologies determined by operator modules and ideal cones --- p.56 / Chapter Chapter 3 --- Banach lattices of operators --- p.63 / Chapter 3.1 --- Cone absolutely summing maps --- p.64 / Chapter 3.2 --- Compact operators on Banach lattices --- p.72 / Chapter 3.3 --- PL-compact operators and locally order precompact operators --- p.85 / Chapter 3.4 --- Almost order bounded sets and semicompact operators --- p.100 / Ref erences --- p.117

The notion of uniform countable additivity or uniform absolute continuity is present implicitly in the Lebesgue Dominated Convergence Theorem and explicitly in the Vitali-Hahn-Saks and Nikodym Theorems, respectively. V. M. Dubrovsky studied the connection between uniform countable additivity and uniform absolute continuity in a series of papers, and Bartle, Dunford, and Schwartz established a close relationship between uniform countable additivity in ca(Σ) and operator theory for the classical continuous function spaces C(K). Numerous authors have worked extensively on extending and generalizing the theorems of the preceding authors. Specifically, we mention Bilyeu and Lewis as well as Brooks and Drewnowski, whose efforts molded the direction and focus of this paper. This paper is a study of the techniques used by Bell, Bilyeu, and Lewis in their paper on uniform exhaustivity and Banach lattices to present a Banach lattice version of two important and powerful results in measure theory by Brooks and Drewnowski. In showing that the notions of exhaustivity and continuity take on familiar forms in certain Banach lattices of measures they show that these important measure theory results follow as corollaries of the generalized Banach lattice versions. This work uses their template to generalize results established by Bator, Bilyeu, and Lewis.

Student Number : 0413210T - MSc dissertation - School of Mathematics - Faculty of Science / The aim of this work is to do a literature study on spaces of martingales on Riesz spaces and Banach lattices, using [16, 19, 20, 17, 18, 2, 30] as a point of departure. Convergence of martingales in the classical theory of stochastic processes has many applications in mathematics and related areas. Operator theoretic approaches to the classical theory of stochastic processes and martingale theory in particular, can be found in, for example, [4, 5, 6, 7, 13, 15, 26, 27]. The classical theory of stochastic processes for scalar-valued measurable functions on a probability space ( ,#6;, μ) utilizes the measure space ( ,#6;, μ), the norm structure of the associated Lp(μ)-spaces as well as the order structure of these spaces. Motivated by the existing operator theoretic approaches to classical stochastic processes, a theory of discrete-time stochastic processes has been developed in [16, 19, 20, 17, 18] on Dedekind complete Riesz spaces with weak order units. This approach is measure-free and utilizes only the order structure of the given Riesz space. Martingale convergence in the Riesz space setting is considered in [18]. It was shown there that the spaces of order bounded martingales and order convergent martingales, on a Dedekind complete Riesz space with a weak order unit, coincide. A measure-free approach to martingale theory on Banach lattices with quasi-interior points has been given in [2]. Here, the groundwork was done to generalize the notion of a filtration on a vector-valued Lp-space to the M-tensor product of a Banach space and a Banach lattice (see [1]). In [30], a measure-free approaches to martingale theory on Banach lattices is given. The main results in [30] show that the space of regular norm bounded martingales and the space of norm bounded martingales on a Banach lattice E are Banach lattices in a natural way provided that, for the former, E is an order continuous Banach lattice, and for the latter, E is a KB-space. The definition of a ”martingale” defined on a particular space depends on the type of space under consideration and on the ”filtration,” which is a sequence of operators defined on the space. Throughout this dissertation, we shall consider Riesz spaces, Riesz spaces with order units, Banach spaces, Banach lattices and Banach lattices with quasi-interior points. Our definition of a ”filtration” will, therefore, be determined by the type of space under consideration and will be adapted to suit the case at hand. In Chapter 2, we consider convergent martingale theory on Riesz spaces. This chapter is based on the theory of martingales and their properties on Dedekind complete Riesz spaces with weak order units, as can be found in [19, 20, 17, 18]. The notion of a ”filtration” in this setting is generalized to Riesz spaces. The space of martingales with respect to a given filtration on a Riesz space is introduced and an ordering defined on this space. The spaces of regular, order bounded, order convergent and generated martingales are introduced and properties of these spaces are considered. In particular, we show that the space of regular martingales defined on a Dedekind complete Riesz space is again a Riesz space. This result, in this context, we believe is new. The contents of Chapter 3 is convergent martingale theory on Banach lattices. We consider the spaces of norm bounded, norm convergent and regular norm bounded martingales on Banach lattices. In [30], filtrations (Tn) on the Banach lattice E which satisfy the condition 1[n=1 R(Tn) = E, where R(Tn) denotes the range of the filtration, are considered. We do not make this assumption in our definition of a filtration (Tn) on a Banach lattice. Our definition yields equality (in fact, a Riesz and isometric isomorphism) between the space of norm convergent martingales and 1Sn=1R(Tn). The aforementioned main results in [30] are also considered in this chapter. All the results pertaining to martingales on Banach spaces in subsections 3.1.1, 3.1.2 and 3.1.3 we believe are new. Chapter 4 is based on the theory of martingales on vector-valued Lp-spaces (cf. [4]), on its extension to the M-tensor product of a Banach space and a Banach lattice as introduced by Chaney in [1] (see also [29]) and on [2]. We consider filtrations on tensor products of Banach lattices and Banach spaces as can be found in [2]. We show that if (Sn) is a filtration on a Banach lattice F and (Tn) is a filtration on a Banach space X, then 1[n=1 R(Tn Sn) = 1[n=1 R(Tn) e M 1[n=1 R(Sn). This yields a distributive property for the space of convergent martingales on the M-tensor product of X and F. We consider the continuous dual of the space of martingales and apply our results to characterize dual Banach spaces with the Radon- Nikod´ym property. We use standard notation and terminology as can be found in standard works on Riesz spaces, Banach spaces and vector-valued Lp-spaces (see [4, 23, 29, 31]). However, for the convenience of the reader, notation and terminology used are included in the Appendix at the end of this work. We hope that this will enhance the pace of readability for those familiar with these standard notions.

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.