Studies on iconvergent sequence spaces

Abstract

INTRODUCTION A study on sequence space was further extended through summability theory. Studies on the summability methods as a class of transformations of complex sequences by complex infinite matrices was introduced by O. Toeplitz. This subject was further studied by G.H. Hardy, I. Schur, S. Mazur, W. Orlicz, K. Knopp, G.M. Peterson, S. Banach, J.A. Fridy 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. Studies on I- convergence was introduced by P. Kostyrko, T. Salat and W. Wilezynski. This subject was further investigated and studied by T. Salat, B.C Tripathy, M. Ziman, M. Et, A. Esi, K. Demirci, S. Yardimci and many others. OBJECTIVE OF THE STUDY The aim of the work is to introduce some I-convergent sequence spaces of complex numbers and study their different topological and algebraic properties. Also establish some inclusion relations involving these sequence spaces and some existing sequence spaces. In the first chapter mention most of the existing definitions and results; those are used in subsequent chapters of the thesis. Also 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. THE SUMMARY OF THE WORK DONE Throughout w, c, c0 and l will represent the classes of all, convergent, null and bounded sequences respectively. Let X be a non empty set. A non empty family of sets I2x (power sets of X) is said to an ideal if I is additive (i.e. A,BEIABEI) and hereditary (i.e. AEI, BA BEI). The notion I-convergence was introduced by P. Kostyrko, T. Salat and W. Wilczynski [69]. A sequence (xk) Ew is said to be I-convergent to the number L if for every Egt0, {kEN :/xk - L/ gt E} EI. We write I-lim xk = L. A sequence (xk) Ew is said to be I-null if L = 0. We write I-lim xk = 0. A sequence (xk) Ew is said to be I- Cauchy if for every Egt0 there exists a number m = m (E) such that {kE N...