“A Study of Topological Structures of Linear Spaces of Generalized Sequences

Abstract

This dissertation deals with the sequence spaces and applications. The various topological and algebraic properties of different sequence spaces defined by Orlicz function have been studied. We introduce and study the sequence spaces that are the generalization of classical sequence spaces of null, convergent, and bounded type. We introduce and study a class c_0 (M,(X,||.||),(a,) ̅α ̅ ) of vector valued difference sequences of null type with the help of Orlicz function. It is the generalization of classical null sequence space. We prove some linear structures and prove some inclusion and equality relations in terms of different parameters a ̅ and α ̅. In the similar fashion, we study the sequence space of bounded type l_∞ (M,(X,||.||),(a,) ̅α ̅ ) of normed space valued difference sequences using Orlicz function M. The containment relations on different parameters are established. The class l_∞ (M,X,(Y,||.||) of Banach space Y - valued functions is introduced as the generalization of bounded complex sequences. The different topological structures have been studied when topologized it with the suitable natural norm. The difference sequence spaces W_0 (∆,f),W(∆,f) and W_∞ (∆,f) defined by non-negative Φ-function on R are introduced and studied their different topological properties endowed by paranormed structure on these spaces. Infinite series and sequences played important role in the development of Calculus and other branches of mathematics. But the mathematicians were facing the problems of calculating the limits of infinite sequences and series, in particular with those having divergent in behaviour. Then the mathematicians developed the various types of convergence to assign a limit in some sense to divergent sequences and series. We also introduce and investigate sequence spaces defined by ideal convergence and Orlicz function in 2-normed space. The theory of sequence space and frame theory are interconnected as frame theory makes the use of sequence space. The sequence spaces are used as the vector spaces in frame theory. Some of the application of frame theory that makes the use of sequence spaces are image processing, signal processing, error correction, data compression etc. The atomic decomposition in a non-locally convex Banach space is defined and discussed its existence. It is also proved that if a p-Banach space has an atomic decomposition then the space is isomorphic to its associated p-Banach sequence space. The necessary and sufficient condition for an atomic decomposition in p-Banach space is given. Certain properties associated with Schauder frames in Banach space have been defined and studied.