Sequence Spaces and Summability over Valued Fields

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. Cover Half Title Title Page Copyright Page Dedication Contents About the Author Foreword Preface 1. Preliminaries 1.1 Valuation and the topology induced by it 1.2 Kinds of valuations 1.3 Normed linear spaces 1.4 K-convexity and locally K-convex spaces 1.5 Topological algebras 1.6 Summability methods 2. On Certain Spaces Containing the Space of Cauchy Sequences 2.1 Introduction 2.2 Summability of sequences of 0's and 1's 2.3 Some structural properties of l∞ 2.4 The Steinhaus theorem 2.5 A Steinhaus-type theorem 3. Matrix Transformations between Some Other Sequence Spaces 3.1 Introduction 3.2 Characterization of matrices in (lα,lα), α>0 3.3 Multiplication of series 3.4 A Mercerian theorem 3.5 Another Steinhaus-type theorem 3.6 Characterization of matrices in (l(p),l∞) 4. Characterization of Regular and Schur Matrices 4.1 Introduction 4.2 Summability of subsequences and rearrangements 4.3 The core of a sequence 5. A Study of the Sequence Space c0(p) 5.1 Identity of weak and strong convergence or the Schur property 5.2 Normability 5.3 Nuclearity of c0(p) 5.4 c0(p) as a Schwartz space 5.5 c0(p) as a metric linear algebra 5.6 Step spaces 5.7 Some more properties of the sequence space c0(p) 6. On the Sequence Spaces l(p), c0(p), c(p), l∞(p) over Non-Archimedean Fields 6.1 Introduction 6.2 Continuous duals and the related matrix transformations 6.3 Some more properties of the sequence spaces l(p),c0(p) and l∞(p) 6.4 On the algebras (c,c) and (lα,lα) 7. A Characterization of the Matrix Class (l∞,c0) and Summa-bility Matrices of Type M in Non-Archimedean Analysis 7.1 Introduction 7.2 A Steinhaus-type theorem 7.3 A characterization of the matrix class (l∞,c0) 7.4 Summability matrices of type M 8. More Steinhaus-Type Theorems over Valued Fields 8.1 Introduction 8.2 A Steinhaus-type theorem when K = R or C 8.3 Some Steinhaus-type theorems over valued elds 8.4 Some more Steinhaus-type theorems over valued fields I 8.5 Some more Steinhaus-type theorems over valued fields II Index The main object of the present book is two-fold. Firstly, it supplements the comparatively small bulk of literature relating to sequence spaces and matrix transformations in non-archimedean analysis. Secondly, it illustrates how new proofs are necessary for proving the exact analogues of classical results.