Double Sequence Spaces and Four-Dimensional Matrices

Description

Double Sequence Spaces and Four-Dimensional Matrices provides readers with a clear introduction to the spaces of double sequences and series, as well as their properties. The book then goes beyond this to investigate paranormed double sequence spaces and their algebraic and topological properties, triangle matrices and their domains in certain spaces of double sequences, dual spaces of double sequence spaces, and matrix transformations between double sequence spaces and related topics. Each chapter contains a conclusion section highlighting the importance of results and pointing out possible new ideas that can be studied further. Features Suitable for students at graduate or post-graduate level and researchers Investigates different types of summable spaces and computes their duals Characterizes several four-dimensional matrix classes transforming one summable space into other Discusses several algebraic and topological properties of new sequence spaces generated by the domain of triangles. Cover Half Title Title Page Copyright Page Dedication Contents Foreword Preface Authors List of Abbreviations and Symbols 1. Spaces of Double Sequences and Series 1.1. Double Sequence Spaces 1.2. Double Series Spaces 1.3. Four-Dimensional Infinite Matrices 1.4. Some Topological Properties of the Spaces Almost Null and Almost Convergent Double Sequences 1.5. Riesz Summability of Double Series 1.6. Abel Summability of Double Series 1.7. Conclusion 1.8. Exercises 2. Some Paranormed Double Sequence Spaces 2.1. Preliminaries, Background and Notations 2.2. The Double Sequence Spaces Cp(t) and Cbp(t) 2.3. Paranormed Space Mu(t) of Double Sequences 2.4. Double Sequence Spaces Cp0(t), Cbp0(t) and Lu(t) 2.5. Paranormed Spaces Cr(t), Cr0(t), Ctr(t) and Ctr0(t) 2.6. Conclusion 2.7. Exercises 3. Matrix Domain in Double Sequence Spaces 3.1. Preliminaries, Background and Notations 3.2. Certain Paranormed Difference Spaces of Double Sequences 3.3. Double Sequence Spaces BS, BS(t), CS# and BV 3.4. Difference Spaces of Almost Convergent and Strongly Almost Convergent Double Sequences 3.5. Absolutely s-Summable Double Sequences 3.6. Riesz Domains in Some Double Sequence Spaces 3.7. Domain of Euler Mean in the Space of Absolutely s-summable Double Sequences With 0 < s < 1 3.8. Conclusion 3.9. Exercises 4. Dual Spaces of Double Sequence Spaces 4.1. Preliminaries, Background and Notations 4.2. Dual Spaces of Double Series 4.3. Dual Spaces of Mu(t) 4.4. Dual Spaces of Cp0(t), Cbp(t) and Cbp0(t) 4.5. Dual Spaces of the Space Lu(t) 4.6. The Alpha-dual of the Space Mu(t) 4.7. Dual Spaces of the Spaces Cf(Δ), Cf0(Δ) and ... 4.8. Dual Spaces of the Spaces Cr(t), Cr0(t), Ctr(t) and Ctr0(t) 4.9. Dual Spaces of the Spaces BVs and LSs 4.10. Dual Spaces of Riesz Spaces Rqt(Mu), Rqt(C#) and Rqt(Ls) 4.11. Dual Spaces of Epsr Spaces 4.12. Conclusion 4.13. Exercises 5. Matrix Transformations Between Double Sequence Spaces 5.1. Preliminaries, Background and Notations 5.2. Characterizations of Some Matrix Classes 5.3. Mercerian and Steinhaus Type Theorems for Four-Dimensional Matrices 5.4. Comparison of Four-Dimensional Summability Methods 5.5. Four-Dimensional Usual Dual Summability Methods 5.6. Four-Dimensional Dual Summability Methods of the New Sort 5.7. Summability of Unbounded Double Sequences 5.8. Some Tauberian Theorems for Four-Dimensional Euler and Borel Summabilities 5.9. Conclusion 5.10. Exercises Bibliography Index