Song-Jian Han. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1994. / Includes bibliographical references (leaves 72-73). / Acknowledgement --- p.i / Abstract --- p.ii / Introduction --- p.iii / Chapter 1 --- Topological Vector Spaces and Elemantary Duality Theory --- p.1 / Chapter 1.1 --- Locally Convex Spaces --- p.2 / Chapter 1.2 --- Bornological Spaces and Bornological Vector Spaces --- p.4 / Chapter 1.3 --- Elementary Properties of Dual Spaces --- p.6 / Chapter 1.4 --- Topological Injections and Surjections Bornological Injections and Surjections --- p.10 / Chapter 2 --- Locally Convex Riesz Spaces --- p.15 / Chapter 2.1 --- Ordered Vector Spaces --- p.15 / Chapter 2.2 --- Riesz Space --- p.18 / Chapter 2.3 --- Locally Convex Riesz Spaces --- p.20 / Chapter 3 --- Half-Full Injections and Half-Decomposable Surjections Half- Full Bornological Injections and Half-Decomposable Bornologi- cal Surjections --- p.24 / Chapter 4 --- Operator Modules between Locally Convex Riesz Spaces --- p.35 / Chapter 4.1 --- Preliminaries --- p.35 / Chapter 4.2 --- Operator Modules and Ideal Cones --- p.37 / Chapter 4.3 --- The Half-Full Injective Hull and the Half-Decomposable Bornolog- ical Surjective Hull of Operator Modules Between Locally Convex Riesz Spaces --- p.41 / Chapter 4.4 --- Extensions of Operator Modules and Ideal Cones --- p.57 / References --- p.72

Chan Ka Wo. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2000. / Includes bibliographical references (leaves 114-118). / Abstracts in English and Chinese. / Introduction --- p.iii / Conventions of This Thesis --- p.vi / Prerequisites --- p.xiii / Chapter 1 --- Cones in Real Vector Spaces --- p.1 / Chapter 1.1 --- The Fundamentals of Cones --- p.2 / Chapter 1.2 --- Enlargements of a Cone --- p.22 / Chapter 1.3 --- Special Cones in Real Vector Spaces --- p.29 / Chapter 1.3.1 --- Positive Cones --- p.29 / Chapter 1.3.2 --- Bishop-Phelps Cones --- p.36 / Chapter 1.3.3 --- Quasi-Bishop-Phelps Cones --- p.42 / Chapter 1.3.4 --- Quasi*-Bishop-Phelps Cones --- p.45 / Chapter 1.3.5 --- Gallagher-Saleh D-cones --- p.47 / Chapter 2 --- Generalizations in Topological Vector Spaces --- p.52 / Chapter 2.1 --- Efficiency and Positive Proper Efficiency --- p.54 / Chapter 2.2 --- Type I Generalizations --- p.71 / Chapter 2.3 --- Type II Generalizations --- p.82 / Chapter 2.4 --- Type III Generalizations --- p.92 / Chapter 3 --- Generalizations in Dual Spaces --- p.97 / Chapter 3.1 --- Weak*-Support Points of a Set --- p.98 / Chapter 3.2 --- Generalizations in the Dual Space of a General Normed Space --- p.100 / Chapter 3.3 --- Generalizations in the Dual Space of a Banach Space --- p.104 / Epilogue: Glimpses Beyond --- p.112 / Bibliography --- p.114

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.

Thesis (Ph.D.)--Ohio University, August, 2009. / Release of full electronic text on OhioLINK has been delayed until September 1, 2012. Includes bibliographical references (leaves 110-113)

Thesis (Ph.D.)--Ohio University, August, 2009. / Title from PDF t.p. Release of full electronic text on OhioLINK has been delayed until September 1, 2012. Includes bibliographical references (leaves 110-113)

published_or_final_version / Mathematics / Doctoral / Doctor of Philosophy

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We present three criteria for compactness in the context of apartness spaces and Bishop-style constructive mathematics. Each of our three criteria can be summarised as requiring that there is a positive distance between any two disjoint closed sets. Neat locatedness and the product apartness give us three variations on this theme. We investigate how our three criteria relate to one another and to several existing compactness criteria, namely classical compactness, completeness, total boundedness, the anti-Specker property, and Diener's neat compactness.

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