Studies on some vector valued sequence spaces and kthe toeplitz duals

Abstract

1. INTRODUCTION Studies on sequence space was further extended through summability theory. The summability theory originated from the attempts made by the mathematicians to give limits to the divergent sequences, on taking its transformations. O. Toeplitz was the first person to study the summability methods as a class of transformations of complex sequences by complex infinite matrices. It was followed by the works due to G.H. Hardy, I. Schur, S Mazur, W. Orlicz, K. Knopp, G.M. Petersen, S. Banach, G. Kothe, J. A. Fridy and T. Pati are a few to be named. The works on paranormed sequence spaces was initiated by H. Nakano and S. Simons. It was further studied by I.J. Maddox, C. G. Lascarides, S. Nanda, D. Rath, G. Das, B. Kuttner and many others. The scope for the studies on sequence spaces was further extended by the application of different techniques and notions of functional analysis. The earlier works on double sequences is found in Bromwich [6]. It was further investigated in the beginning of nineteenth century by the eminent mathematician G.H. Hardy. Later on it was further investigated and studied from summability theory by F. Moricz, B.E. Rhoades, O. Sonalcan, B.C. Tripathy and many others. 2. OBJECTIVE OF THE STUDY The aim of the work carried is to introduce some new vector valued sequence spaces and study their different properties like completeness, solidity, separability, symmetricity etc. Further inclusion relations involving the introduced sequence spaces and some existing sequence spaces have been established. The first chapter of the thesis is introductory in nature. Most of the existing definitions and results have been collected in this chapter, those are used in subsequent chapters of the thesis. Further the preliminaries of the works carried are given to have a clear picture of the background and the development of the topics on which the works have been carried in this thesis. 3. THE SUMMARY OF THE WORK DONE An Orlicz function M is a mapping M: [0,)[0,) such that it is continuous,...