Sequence spaces and summability over valued fields is a research book aimed at research scholars, graduate students and teachers with an interest in Summability Theory both Classical (Archimedean) and Ultrametric (non-Archimedean).
The book presents theory and methods in the chosen topic, spread over 8 chapters that seem to be important at research level in a still developing topic.
Key Features
- Presented in a self-contained manner
- Provides examples and counterexamples in the relevant contexts
- Provides extensive references at the end of each chapter to enable the reader to do further research in the topic
- Presented in the same book, a comparative study of Archimedean and non-Archimedean Summability Theory
- Appeals to young researchers and experienced mathematicians who wish to explore new areas in Summability Theory
The book is written by a very experienced educator and researcher in Mathematical Analysis particularly Summability Theory.
| About the Author |
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xi | |
| Foreword |
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xiii | |
| Preface |
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xvii | |
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1 | (18) |
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1.1 Valuation and the topology induced by it |
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1 | (2) |
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3 | (3) |
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6 | (3) |
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1.4 AT-convexity and locally K-convex spaces |
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9 | (2) |
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11 | (1) |
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12 | (7) |
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2 On Certain Spaces Containing the Space of Cauchy Sequences |
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19 | (36) |
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19 | (3) |
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2.2 Summability of sequences of 0's and 1's |
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22 | (4) |
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2.3 Some structural properties of ∞ |
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26 | (14) |
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2.4 The Steinhaus theorem |
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40 | (10) |
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2.5 A Steinhaus-type theorem |
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50 | (5) |
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3 Matrix Transformations between Some Other Sequence Spaces |
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55 | (24) |
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55 | (1) |
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3.2 Characterization of matrices in (α, α), α > 0 |
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56 | (4) |
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3.3 Multiplication of series |
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60 | (3) |
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63 | (3) |
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3.5 Another Steinhaus-type theorem |
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66 | (5) |
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3.6 Characterization of matrices in ((p), ∞) |
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71 | (8) |
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4 Characterization of Regular and Schur Matrices |
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79 | (18) |
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79 | (2) |
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4.2 Summability of subsequences and rearrangements |
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81 | (8) |
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4.3 The core of a sequence |
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89 | (8) |
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5 A Study of the Sequence Space c0(p) |
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97 | (32) |
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5.1 Identity of weak and strong convergence or the Schur property |
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97 | (5) |
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102 | (2) |
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104 | (6) |
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5.4 c0(p) as a Schwartz space |
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110 | (7) |
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5.5 c0(p) as a metric linear algebra |
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117 | (4) |
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121 | (1) |
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5.7 Some more properties of the sequence space C0(p) |
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121 | (8) |
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6 On the Sequence Spaces (p), c0(p), c(p), ∞(p) over Non-Archimedean Fields |
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129 | (22) |
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129 | (1) |
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6.2 Continuous duals and the related matrix transformations |
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130 | (6) |
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6.3 Some more properties of the sequence spaces (p), c0(p), c(p), ∞(p) |
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136 | (7) |
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6.4 On the algebras (c, c) and (α, α) |
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143 | (8) |
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7 A Characterization of the Matrix Class (∞, c0) and Summability Matrices of Type M in Non-Archimedean Analysis |
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151 | (16) |
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151 | (1) |
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7.2 A Steinhaus-type theorem |
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152 | (1) |
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7.3 A characterization of the matrix class (∞, c0) |
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153 | (4) |
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7.4 Summability matrices of type M |
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157 | (10) |
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8 More Steinhaus-Type Theorems over Valued Fields |
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167 | (22) |
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167 | (1) |
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8.2 A Steinhaus-type theorem when K = R or C |
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167 | (5) |
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8.3 Some Steinhaus-type theorems over valued fields |
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172 | (4) |
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8.4 Some more Steinhaus-type theorems over valued fields I |
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176 | (5) |
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8.5 Some more Steinhaus-type theorems over valued fields II |
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181 | (8) |
| Index |
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189 | |
P.N. Natarajan is former Professor and Head, Department of Mathematics, Ramakrishna Mission Vivekananda College, Chennai. He did his Ph.D. from the University of Madras, under Prof. M.S. Rangachari, Former Director and Head, The Ramanujan Institute for Advanced Study in Mathematics, University of Madras. An active researcher, Prof. Natarajan has over 100 research papers to his credit published in International Journals like Proceedings of the American Mathematical Society, Bulletin of the London Mathematical Society, Indagationes Mathematicae, Annales Mathematiques Blaise Pascal and Commentationes Mathematicae. He has so far written 4 books and contributed a chapter each to two edited volumes, published by the famous International publishers Springer, Taylor-Francis and Wiley. His research interest includes Summability Theory and Functional Analysis, both Classical and Ultrametric. Prof. Natarajan was honoured with the Dr. Radhakrishnan Award for the Best Teacher in Mathematics for the 1990-1991 by the Government of Tamil Nadu. Besides visiting several institutes of repute in Canada, France, Holland and Greece on invitation, Prof. Natarajan has participated in several International Conferences and has chaired sessions.
