The purpose of this paper is to explore certain properties of Banach spaces. The first chapter begins with basic definitions, includes examples of Banach spaces, and concludes with some properties of continuous linear functionals. In the second chapter, dimension is discussed; then one version of the Hahn-Banach Theorem is presented. The third chapter focuses on dual spaces and includes an example using co, RI, and e'. The role of locally convex spaces is also explored in this chapter. In the fourth chapter, several more theorems concerning dual spaces and related topologies are presented. The final chapter focuses on reflexive spaces. In the main theorem, the relation between compactness and reflexivity is examined. The paper concludes with an example of a non-reflexive space.

Ph.D. (Mathematics) / Our objective in this thesis is to investigate two fundamental questions concerning Riesz operators de ned on a Banach space. Recall that Riesz operators are generalizations of compact operators in the sense that Riesz operators have the same spectral properties as compact operators. However, they do not possess the same algebraic properties as compact operators. Our rst question that we investigate is: When is a Riesz operator a nite rank operator? This question is motivated from the fact that if a compact operator de ned on a Banach space has closed range, then it is a nite rank operator. Also, Ghahramani proved that a compact homomorphism de ned on a C -algebra is a nite rank operator, see . Martin Mathieu, in his paper, generalized the result of Ghahramani by proving that a weakly compact homomorphism de ned on a C -algebra is a nite rank operator...

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The purpose of this paper is to present the results of a study of linear spaces with special emphasis of linear transformations, norms, and inner products.

Researchers continue to emphasize the important role supervisors have in creating a safe space for supervisees to effectively navigate and engage in honest multicultural conversations while also addressing potential biases (e.g., Ancis & Marshall, 2010). However, much of the literature on multicultural supervision provides limited guidance on what characteristics define a safe space. The purpose of this study was to examine accounts of counseling psychology graduate students to learn about their understanding and definitions of safe space and brave space within multicultural supervision, using grounded theory methodology (Corbin & Strauss, 2015). Specifically, the purpose of the study was threefold: (a) to define what a safe space and a brave space is from supervisees’ perspective, (b) to determine if the concept of a safe space is viewed similarly or differently to a brave space, and (c) to identify specific behaviors and interventions that supervisors perform that make a supervisee feel they are in a safe or brave space. Results yielded a model characterized by three core dimensions that comprise safe and brave spaces including: (a) safety within the physical space, (b), definitions and use of brave and safe spaces, and (c) supervisor actions and behaviors. This study adds to the current multicultural supervision training scholarship by providing new perspectives on how supervisees in counseling psychology doctoral programs make sense of safe and brave spaces during multicultural supervision and what supervisors can do to create such spaces.

Bibliography: pages 90-94. / Ordered spaces were introduced by Leopoldo Nachbin [1948 a, b, c, 1950, 1965]. We will be primarily concerned with completely regular ordered spaces, because they are precisely those ordered spaces which admit quasi-uniform structures. A recent and convenient study of these spaces is in the book by P. Fletcher and W.F. Lindgren [1982]. In this thesis we consider functorial quasi-uniformities over (partially) ordered spaces. The functorial methods which we use were developed by Brummer [1971, 1977, 1979, 1982] and Brummer and Hager [1984, 1987] in the context of functorial uniformities over completely regular topological spaces, and of functorial quasi-uniformities over pairwise. completely regular bitopological spaces. We obtain results which are to a large extent analogous to results in those papers. We also introduce some functors which relate our functorial quasi-uniformities to the structures studied by Brummer and others (e.g. Salbany [1984]).

The thesis is an investigation of literary spaces and relationships.  How could the words from a novel translate into Architecture by using imagination and memories. Could the characters of imagination give a site or a space a certain atmosphere or create certain fragments of memories.

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<p> We look at methods to generate uniformly distributed points from the classical matrix groups, spheres, projective spaces, and Grassmannians. We motivate the discussion with a number of applications ranging from number theory to wireless communications. The uniformity of the samples and the efficiency of the algorithms are compared. </p> / Thesis / Master of Science (MSc)