Sequence spaces

This Research Note presents a collection of topics on sequence spaces and infinite matrices. It contains discussions of Kothe sequence spaces, topologies on sequence spaces, matrix mappings between sequence spaces and topics from summability theory. The development is original and includes new proofs of Pitt's theorem and the bounded consistency theorem, and previously unpublished results on matrix mappings between Kothe spaces having their normal topologies. The work will be of particular interest to researchers and graduate students in functional analysis and summability theory, and workers in discrete system theory. -- Back cover. Read more... Introduction, basic properties -- Kothe sequence spaces -- Topologies on sequence spaces -- Mappings between sequence spaces -- Topics from summability theory Cover......Page 1 Title: Sequence spaces......Page 2 ISBN 0-273.-08507-7......Page 3 Preface......Page 4 Contents......Page 6 1. ORIENTATION, CLASSIFICATION OF TYPICAL PROBLEMS......Page 8 1.5 Assignment of topologies.......Page 9 2. EXAMPLES OF SEQUENCE SPACES......Page 10 Exercises......Page 12 3. ALGEBRAIC AND COMBINATORIAL PROPERTIES......Page 13 Exercises......Page 15 4. GALOIS DUALITY THEORY WITH RESPECT TO PRODUCT AND CONVOLUTION......Page 16 Exercises......Page 19 5. METHODS OF GENERATING SEQUENCE SPACES......Page 20 Exercises......Page 24 NOTES ON CHAPTER ONE......Page 25 1. INTRODUCTION......Page 27 Exercises......Page 34 2. THE NORMAL TOPOLOGY ON KOTHE SPACES......Page 36 3. SCHWARTZ AND NUCLEAR SEQUENCE SPACES......Page 41 Exercises......Page 49 4. MACKEY AND STRONG TOPOLOGIES......Page 51 Exercises......Page 57 5. REPRESENTATION OF FUNCTION SPACES......Page 58 1. A GENERALIZATION OF THE KOTHE-TOEPLITZ DUALITY THEORY......Page 64 2. THE ZELLER THEORY OF FK-SPACES......Page 66 Exercises......Page 71 3. phi-DUALITY THEORY......Page 72 Exercises......Page 75 4. COMPARISON OF THE GENERALIZED KOTHE-TOEPLITZ THEORY WITH THE phi-DUALITY THEORY. SUM SPACES......Page 76 5. LATTICES OF LOCALLY CONVEX K-SPACES......Page 82 Exercises......Page 89 6. FURTHER LATTICE PROPERTIES OF LOCALLY CONVEX K-SPACES......Page 90 7. ABELIAN AND TAUBERIAN THEOREMS......Page 94 Exercises......Page 101 NOTES ON CHAPTER THREE......Page 102 1. MATRIX MAPPINGS BETWEEN CERTAIN SPACES, CONTINUITY......Page 103 2. THE PREIMAGE OF A MATRIX, ZK-SPAC......Page 108 The Theorems of Silverman-Toeplitz and Kojma-Schur......Page 111 3.3 Theorem (Silverman-Toeplitz......Page 112 Exercise......Page 114 4. MAPPINGS FROM L^1, bvo , bv; THE THEOREM OF LORENTZ AND KNOPP......Page 117 5. MATRIX MAPPINGS FROM l^2 INTO l^2......Page 118 6. DIAGONALS OF MATRIX MAPPINGS......Page 123 7. THE MULTIPLIER ALGEBRA OF A SEQUENCE SPACE......Page 129 8. THE MULTIPLIER ALGEBRA m......Page 140 9. MAPPINGS FROM l^P INTO l^q (p > q): A THEOREM OF PITT......Page 145 Exercises......Page 147 10. THE SHIFT OPERATOR AND CONVOLUTION......Page 148 11. SOME RESULTS ON (S -* T) AND M* (S), AN APPLICATION TO DIFFERENCE EQUATIONS......Page 151 Exercises......Page 156 12. MATRIX MAPPINGS OF KOTHE SPACES......Page 157 NOTES ON CHAPTER FOUR......Page 164 1. SOME PROBLEMS OF SUMMABILITY THEORY......Page 166 2. NORLUND AND CESARO MEANS......Page 168 3. HAUSDORFF MEANS......Page 174 Exercises......Page 180 4. TOPOLOGICAL PROPERTIES OF cA......Page 181 5. THE EXTENT OF cA......Page 186 Exercises......Page 196 6. THE BOUNDED CONSISTENCY THEOREM......Page 197 NOTES ON CHAPTER 5......Page 201 References......Page 202 This Research Note Presents A Collection Of Topics On Sequence Spaces And Infinite Matrices. It Contains Discussions Of Köthe Sequence Spaces, Topologies On Sequence Spaces, Matrix Mappings Between Sequence Spaces And Topics From Summability Theory. The Development Is Original And Includes New Proofs Of Pitt's Theorem And The Bounded Consistency Theorem, And Previously Unpublished Results On Matrix Mappings Between Köthe Spaces Having Their Normal Topologies. The Work Will Be Of Particular Interest To Researchers And Graduate Students In Functional Analysis And Summability Theory, And Workers In Discrete System Theory. -- Back Cover. Introduction, Basic Properties -- Köthe Sequence Spaces -- Topologies On Sequence Spaces -- Mappings Between Sequence Spaces -- Topics From Summability Theory. William H. Ruckle. Pitman Advanced Publishing Program. Includes Bibliographical References.