by Ho Wing Man. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1997. / Includes bibliographical references (leaves 85). / Introduction --- p.1 / Chapter 1 --- Preliminaries --- p.3 / Chapter 1.1 --- Gateaux and Frechet Differentiability --- p.4 / Chapter 1.2 --- β-Differentiability --- p.9 / Chapter 1.3 --- Monotone Operators and Usco Maps --- p.14 / Chapter 2 --- Variational Principle --- p.25 / Chapter 2.1 --- A Generalized Variational Principle --- p.27 / Chapter 2.2 --- A Smooth Variational Principle --- p.37 / Chapter 3 --- Differentiability of Convex Functions --- p.47 / Chapter 3.1 --- On Banach Spaces with β-Smooth Bump Functions --- p.48 / Chapter 3.2 --- A Characterization of Asplund Spaces --- p.64 / Chapter 4 --- More on Differentiability --- p.70 / Chapter 4.1 --- Introduction --- p.70 / Chapter 4.2 --- Differentiability Theorems --- p.75

The central idea in this thesis is the introduction of a new isometric invariant of a Banach space. This is Property AI-I. A Banach space has Property AI-I if whenever a finite metric space almost-isometrically embeds into the space, it isometrically embeds. To study this property we introduce two further properties that can be thought of as finite metric variants of Dvoretzky's Theorem and Krivine's Theorem. We say that a Banach space satisfies the Finite Isometric Dvoretzky Property (FIDP) if it contains every finite subset of $\ell_2$ isometrically. We say that a Banach space has the Finite Isometric Krivine Property (FIKP) if whenever $\ell_p$ is finitely representable in the space then it contains every subset of $\ell_p$ isometrically. We show that every infinite-dimensional Banach space \emph{nearly} has FIDP and every Banach space nearly has FIKP. We then use convexity arguments to demonstrate that not every Banach space has FIKP, and thus we can exhibit classes of Banach spaces that fail to have Property AI-I. The methods used break down when one attempts to prove that there is a Banach space without FIDP and we conjecture that every infinite-dimensional Banach space has Property FIDP.

Ng, Kwong Wing. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2010. / Includes bibliographical references (p. 123-126). / Abstracts in English and Chinese. / Abstract --- p.i / Acknowledgments --- p.iii / Contents --- p.v / Introduction --- p.vii / Chapter 1 --- Preliminaries --- p.1 / Chapter 1.1 --- Basic Notations and Conventions --- p.1 / Chapter 1.2 --- Fundamental Results in Banach Space Theory and Variational Analysis --- p.4 / Chapter 1.3 --- Set-Valued Mappings --- p.6 / Chapter 1.4 --- Enlargements and Projections --- p.8 / Chapter 1.5 --- Subdifferentials --- p.11 / Chapter 1.6 --- Sets of Normals --- p.18 / Chapter 1.7 --- Coderivatives --- p.24 / Chapter 2 --- The Generalized Distance Function - Basic Estimates --- p.27 / Chapter 2.1 --- Elementary Properties of the Generalized Distance Function --- p.27 / Chapter 2.2 --- Frechet-Like Subdifferentials of the Generalized Distance Function --- p.32 / Chapter 2.3 --- Limiting and Singular Subdifferentials of the Generalized Distance - Function --- p.44 / Chapter 3 --- The Generalized Distance Function - Estimates via Intermediate Points --- p.73 / Chapter 3.1 --- Frechet-Like and Limiting Subdifferentials of the Generalized Dis- tance Function via Intermediate Points --- p.74 / Chapter 3.2 --- Frechet and Proximal Subdifferentials of the Generalized Distance Function via Intermediate Points --- p.90 / Chapter 4 --- The Marginal Function --- p.95 / Chapter 4.1 --- Singular Subdifferentials of the Marginal Function --- p.95 / Chapter 4.2 --- Singular Subdifferentials of the Generalized Marginal Function . . --- p.102 / Chapter 5 --- The Perturbed Distance Function --- p.107 / Chapter 5.1 --- Elementary Properties of the Perturbed Distance Function --- p.107 / Chapter 5.2 --- The Convex Case - Subdifferentials of the Perturbed Distance Function --- p.111 / Chapter 5.3 --- The Nonconvex Case - Frechet-Like and Proximal Subdifferentials of the Perturbed Distance Function --- p.113 / Bibliography --- p.123

Dissertation (Ph.D.)--Auburn University, 2008. / Abstract. Includes bibliographic references (p.73-80).

Thesis (Ph. D.)--University of Texas at Austin, 2001. / Vita. Includes bibliographical references. Available also from UMI/Dissertation Abstracts International.

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We introduce the class AUMD of Banach spaces X for which X-valued analytic martingales converge unconditionally. We shew that various possible definitions of this class are equivalent by methods of martingale decomposition. We shew that such X have finite cotype and are q-complex uniformly convex in the sense of Garling. Using multipliers we shew that analytic martingales valued in L¹ converge unconditionally and that AUMD spaces have the analytic Radon-Nikodym property. We shew that X has the AUMD property if and only if strong Hbrmander-Mihlin multipliers are bounded on the Hardy space H¹_x(T). We achieve this by representing multipliers as martingale transforms. It is shewn that if X is in AUMD and is of cotype two then X has the Paley Theorem property. Using an isomorphism result we shew that if A is an injective operator system on a separable Hilbert space and P a completely bounded projection on A, then either PA or (I-P)A is completely boundedly isomorphic to A. The finite-dimensional version of this result is deduced from Ramsey's Theorem. It is shewn that B(e² is primary. It is shewn that weakly compact homomorphisms T from the 2 disc algebra into B(e² are necessarily compact. An explicit form for such T is obtained using spectral projections and it is deduced that such T are absolutely summing.

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Our aim in this research was to study monotone operators in Banach spaces. In particular, the most important concept in this theory, the maximal monotone operators. Here we make an attempt to describe most of the important results and concepts on maximal monotone operators and how they all tie together. We will take a brief look at subdifferentials, which generalize the notion of a derivative. The subdifferential is a maximal monotone operator and it has proved to be of fundamental importance for the study of maximal monotone operators. The theory of maximal monotone operators is somewhat complete in reflexive Banach spaces. However, in nonreflexive Banach spaces it is still to be developed fully. As such, here we will describe most of the important results about maximal monotone operators in Banach spaces and we will distinguish between the reflexive Banach spaces and nonreflexive Banach spaces when a property is known to hold only in reflexive Banach spaces. In the latter case, we will state what the corresponding situation is in nonreflexive Banach spaces and we will give counter examples whenever such a result is known to fail in nonreflexive Banach spaces. The representations of monotone operators by convex functions have found to be extremely useful for the study of maximal monotone operators and it has generated a lot of interest of late. We will discuss some of those key representations and their properties. We will also demonstrate how these representations could be utilized to obtain results about maximal monotone operators. We have included a discussion about the very important Rockafellar sum theorem and some its generalizations. This key result and its generalizations have only been proved in reflexive Banach spaces. We will also discuss several special cases where the Rockafellar sum theorem is known to be true in nonreflexive Banach spaces. The subclasses which provide a basis for the study of monotone operators in nonreflexive Banach spaces are also discussed here