Sequence Spaces Topics in Modern Summability Theory 1st Edition by Mohammad Mursaleen, Feyzi Başar 1000045178 9781000045178

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ISBN-10 :  1000045178 

ISBN-13 :  9781000045178

Author:  Mohammad Mursaleen, Feyzi Başar 

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.

 

Sequence Spaces-Topics in Modern Summability Theory 1st Table of contents:

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 lp (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
5.2.3 Goldberg’s Classification of Spectrum
5.3 The Fine Spectrum of the Operator Defined by the Matrix Λ over the Spaces of Null and Convergent Sequences
5.3.1 The Fine Spectrum of the Operator Λ on the Sequence Space c0
5.3.2 The Fine Spectrum of the Operator Λ on the Sequence Space c
5.4 On the Fine Spectrum of the Upper Triangle Double Band Matrix Δ+ on the Sequence Space c0
5.4.1 The Spectrum and the Fine Spectrum of the Upper Triangle Double Band Matrix Δ+ on the Sequence Space c0
5.5 On the Fine Spectrum of the Generalized Difference Operator Defined by a Double Sequential Band Matrix over the Sequence Space lp, (1 < p < ∞)
5.5.1 The Fine Spectrum of the Operator B(r; s) on the Sequence Space lp
5.6 Fine Spectrum of the Generalized Difference Operator Δuv on the Sequence Space l1
5.6.1 Introduction
5.6.2 Spectrum and Point Spectrum of the Operator Δuv on the Sequence Space l1
5.6.3 Residual and Continuous Spectrum of the Operator Δuv on the Sequence Space l1
5.6.4 Fine Spectrum of the Operator Δuv on the Sequence Space l1
5.6.5 Conclusion
6. Sets of Fuzzy Valued Sequences and Series
6.1 Introduction
6.2 Preliminaries, Background and Notations
6.2.1 Generalized Hukuhara Difference
6.3 Series and Sequences of Fuzzy Numbers
6.3.1 Convergence of the Series of Fuzzy Numbers
6.3.2 The Convergence Tests for the Series of Fuzzy Numbers with Positive Terms
6.4 Power Series of Fuzzy Numbers
6.4.1 Power Series of Fuzzy Numbers with Real or Fuzzy Coefficients
6.5 Alternating and Binomial Series of Fuzzy Numbers with the Level Sets
6.5.1 Fuzzy Alternating Series
6.5.2 Fuzzy Binomial Identity
6.5.3 Examples on the Radius of Convergence of Fuzzy Power Series
6.5.4 Differentiation of Fuzzy Power Series
6.6 On Fourier Series of Fuzzy-Valued Functions
6.6.1 Fuzzy-Valued Functions with the Level Sets
6.6.2 Generalized Hukuhara Differentiation
6.6.3 Generalized Fuzzy-Henstock Integration
6.6.4 Fourier Series for Fuzzy-Valued Functions of Period 2π
6.7 On the Slowly Decreasing Sequences of Fuzzy Numbers
6.7.1 The Main Results
6.8 Determination of the Duals of Classical Sets of Sequences of Fuzzy Numbers and Related Matrix Transformations
6.8.1 Introduction
6.8.2 Determination of Duals of the Classical Sets of Sequences of Fuzzy Numbers
6.8.3 Matrix Transformations Between Some Sets of Sequences of Fuzzy Numbers
6.9 On Some Sets of Fuzzy-Valued Sequences with the Level Sets
6.9.1 Completeness of the Sets of Bounded, Convergent and Null Series of Fuzzy Numbers with the Level Sets
6.9.2 The Duals of the Sets of Sequences of Fuzzy Numbers with the Level Sets
6.10 Certain Sequence Spaces of Fuzzy Numbers Defined by a Modulus Function
6.10.1 The Spaces l∞(F; f), c(F; f), c0(F; f) and lp(F; f; s) of Sequences of Fuzzy Numbers Defined by a 

 

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Tags: Sequence Spaces, Topics, Modern Summability, Mohammad Mursaleen, Feyzi Başar