Geometry of normed linear spaces in light of Birkhoff-James orthogonality

Abstract

In this thesis, we have tried to understand the geometry of normed spaces in the light of Birkhoff-James orthogonality. Using the first two chapters to give introductory notes and to establish relevant notations and terminologies, we have presented the work done in the following four chapters. In chapter three, we have studied the geometry of the normed algebra of holomorphic maps in a neighborhood of a curve and have established a relationship between the geometry of the algebra and the zeros of holomorphic maps. In the fourth chapter, we have studied Birkhoff-James orthogonality and its pointwise symmetry in Lebesgue spaces defined on arbitrary measure spaces and natural numbers. We have found the onto isometries of the sequence spaces using the symmetry of the aforementioned orthogonality. In the fifth chapter, we have studied the geometry of tensor product spaces and have used the results to study the relationship between the symmetry of orthogonality and the geometry (for example, extreme points) of certain spaces of operators. Our work in this chapter is motivated by the famous Grothendieck inequality. In the sixth and final chapter, we have studied the geometry of ℓp and c0 direct sums of normed spaces (1 ⩽ p <∞). We characterize the smoothness and approximate smoothness of these spaces along with Birkhoff-James orthogonality and its pointwise symmetry. As a consequence of our study, we have answered a question pertaining to the approximate smoothness of a space raised by Chmeili´enski, Khurana, and Sain in [14].