Sequence spaces : topics in modern summability theory / Mohammad Mursaleen, Feyzi Basar.

This book is aimed at both experts and non-experts with an interest in getting acquainted with sequence spaces, matrix transformations and their applications. It consists of several new results which are part of the recent research on these topics. It provides different points of view in one volume, e.g. their topological properties, geometry and summability, fuzzy valued study and more. This book presents the important role sequences and series play in everyday life, it covers geometry of Banach Sequence Spaces, it discusses the importance of generalized limit, it offers spectrum and fine spectrum of several linear operators and includes fuzzy valued sequences which exhibits the study of sequence spaces in fuzzy settings. This book is the main attraction for those who work in Sequence Spaces, Summability Theory and would also serve as a good source of reference for those involved with any topic of Real or Functional Analysis.

Cover -- Half Title -- Series Page -- Title Page -- Copyright Page -- Contents -- Foreword -- Preface -- Acknowledgements -- Authors -- List of Abbreviations and Symbols -- 1. Basic Functional Analysis -- 1.1 Metric Spaces -- 1.2 Metric Sequence Spaces -- 1.3 Normed Linear Spaces -- 1.4 Bounded Linear Operators -- 1.5 Köthe-Toeplitz Duals -- 1.6 Basic Theorems -- 1.7 Compact Operators -- 1.8 Schauder Basis and Separability -- 1.9 Reflexivity -- 1.10 Weak Convergence -- 1.11 Hilbert Spaces -- 1.12 Topological Vector Spaces -- 1.13 Linear Metric Spaces -- 1.14 FK-Spaces

2. Geometric Properties of Some Sequence Spaces -- 2.1 Introduction -- 2.2 Geometric Properties -- 2.3 Orlicz Sequence Spaces -- 2.4 Cesàro Sequence Spaces -- 2.5 Sequence Spaces Related to lp Spaces -- 2.6 Sequence Spaces lp(u, v) and l (u, v, p) -- 3. Infinite Matrices -- 3.1 Introduction -- 3.2 Matrix Transformations Between Some FK-Spaces -- 3.3 Conservative Matrices -- 3.4 Schur Matrices -- 3.5 Examples of Regular Matrices -- 3.5.1 Cesàro Matrix -- 3.5.2 Euler Matrix -- 3.5.3 Riesz Matrix -- 3.5.4 Nörlund Matrix -- 3.5.5 Borel Matrix -- 3.5.6 Abel Matrix

4. Almost Convergence and Classes of Related Matrix Transformations -- 4.1 Introduction -- 4.2 Almost Convergence -- 4.3 Almost Conservative and Almost Regular Matrices -- 4.4 Almost Coercive Matrices -- 4.5 Strongly Regular Matrices -- 4.6 Applications to Approximation -- 5. Spectrum of Some Triangle Matrices on Some Sequence Spaces -- 5.1 Preliminaries, Background and Notations -- 5.2 Subdivisions of Spectrum -- 5.2.1 The Point Spectrum, Continuous Spectrum and Residual Spectrum -- 5.2.2 The Approximate Point Spectrum, Defect Spectrum and Compression Spectrum

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