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Thesis (Ph.D.)--Ohio University, June, 2009. / Includes bibliographical references (leaves 146-149)

Thesis (Ph.D.)--Ohio University, June, 2009. / Title from PDF t.p. Includes bibliographical references (leaves 146-149)

Metric topologies can be viewed as one-dimensional measures. This dissertation is a topological study of two-dimensional measures. Attention is focused on locally convex vector topologies on infinite dimensional real spaces. A nabla (referred to in the literature as a 2-norm) is the analogue of a norm which assigns areas to the parallelograms. Nablas are defined for the classical normed spaces and techniques are developed for defining nablas on arbitrary spaces. The work here brings out a strong connection with tensor and wedge products. Aside from the normable theory, it is shown that nabla topologies need not be metrizable or Mackey. A class of concretely given non-Mackey nablas on the ℓp and Lp spaces is introduced and extensively analyzed. Among other results it is found that the topological dual of ℓ₁ with respect to these nabla topologies is C₀, one of the spaces infamous for having no normed predual. Also, a connection is made with the theory of two-norm convergence (not to be confused with 2-norms). In addition to the hard analysis on the classical spaces, a duality framework from which to study the softer aspects is introduced. This theory is developed in analogy with polar duality. The ideas corresponding to barrelledness, quasi-barrelledness, equicontinuity and so on are developed. This dissertation concludes with a discussion of angles in arbitrary normed spaces and a list of open questions.

In this paper the Cartesian product topology for an arbitrary family of topological spaces and some of its basic properties are defined. The space is investigated to determine which of the separation properties of the component spaces are invariant.

If X is a topological space and Y is a ring of closed sets, then a necessary and sufficient condition for the Wallman space W(X,F) to be a compactification of X is that X be T1 andYF separating. A necessary and sufficient condition for a Wallman compactification to be Hausdoff is that F be a normal base. As a result, not all T, compactifications can be of Wallman type. One point and finite Hausdorff compactifications are of Wallman type.

The goal of this paper is to characterize, at least partially, upper semi-continuous decompositions of topological spaces and the role that upper semi-continuity plays in preserving certain topological properties under decomposition mappings. Attention is also given to establishing what role upper semi-continuity plays in determining conditions under which decomposition spaces possess certain properties. A number of results for non-upper semi-continuous decompositions are included to help clarify the scope of the part upper semi-continuity plays in determining relationships between topological spaces and their decomposition spaces.

A research report submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in partial fulfilment of the requirements for the degree of Master of Science. / This research report serves as an introduction to the concept of Operator Spaces which has gained considerable momentum in its acknowledgement and research interest in the last few decades. It will highlight a very important breakthrough on the characterization of Operator spaces which occurred in the !ast few years brought about by Z.J. Ruan. It investigates the relationship of this space in relation to Banach space theory by looking at an extension theorem for linear functionals, / Andrew Chakane 2018

by Yun Chung Chu. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1992. / Includes bibliographical references (leaves 163-[165]). / Chapter 1 --- Background --- p.1 / Chapter 1.1 --- Introduction --- p.1 / Chapter 1.2 --- Basic notations --- p.4 / Chapter 1.3 --- Object of study --- p.6 / Chapter 2 --- Review of Kohonen's algorithm --- p.22 / Chapter 2.1 --- General form of Kohonen's algorithm --- p.22 / Chapter 2.2 --- r-neighborhood by matrix --- p.25 / Chapter 2.3 --- Examples --- p.28 / Chapter 3 --- Local minima --- p.34 / Chapter 3.1 --- Theory of local minima --- p.35 / Chapter 3.2 --- Minimizing the number of local minima --- p.40 / Chapter 3.3 --- Detecting the success or failure of Kohonen's algorithm --- p.48 / Chapter 3.4 --- Local minima for different graph structures --- p.59 / Chapter 3.5 --- Formulation by geodesic distance --- p.65 / Chapter 3.6 --- Local minima and Voronoi regions --- p.67 / Chapter 4 --- Induced graph --- p.70 / Chapter 4.1 --- Formalism --- p.71 / Chapter 4.2 --- Practical way to find the induced graph --- p.88 / Chapter 4.3 --- Some examples --- p.95 / Chapter 5 --- Given mapping vs induced mapping --- p.102 / Chapter 5.1 --- Comparison between given mapping and induced mapping --- p.102 / Chapter 5.2 --- Matching the induced mapping to given mapping --- p.115 / Chapter 6 --- A special topic: application to determination of dimension --- p.131 / Chapter 6.1 --- Theory --- p.133 / Chapter 6.2 --- Advanced examples --- p.151 / Chapter 6.3 --- Special applications --- p.156 / Chapter 7 --- Conclusion --- p.159 / Bibliography --- p.163