Infinite Matrices and Sequence Spaces

by Richard G. Cooke

Overview

This clear and correct summation of basic results from a specialized field focuses on the behavior of infinite matrices in general, rather than on properties of special matrices. Three introductory chapters guide students to the manipulation of infinite matrices, covering definitions and preliminary ideas, reciprocals of infinite matrices, and linear equations involving infinite matrices.
From the fourth chapter onward, the author treats the application of infinite matrices to the summability of divergent sequences and series from various points of view. Topics include consistency, mutual consistency, and absolute equivalence; the core of a sequence; the inefficiency and the efficiency problems for infinite matrices; Hilbert vector space and Hilbert matrices; and projective and distance convergence and limit in sequence spaces. Each chapter concludes with examples — nearly 200 in all.

Product Details

ISBN-13:	9780486795065
Publisher:	Dover Publications
Publication date:	05/15/2014
Series:	Dover Books on Mathematics
Sold by:	Barnes & Noble
Format:	eBook
Pages:	368
File size:	25 MB
Note:	This product may take a few minutes to download.

Table of Contents

Chapter 1 Definitions and Preliminary Ideas

1.1 Differences between Finite, and Infinite, Matrix Theory 1

1.2 Some Problems involving the Use of Infinite Matrices 2

1.3 Some Fundamental Definitions 4

1.4 A Few Characteristic Properties of Infinite Matrices 7

1.5 A Few Special Matrices 10

1.6 The Structure of a Matrix 12

1.7 The Exponential Function of an Infinite Lower Semi-matrix 13

1.8 Semi-continuous and Continuous Matrices 15

Examples 1 16

Chapter 2 Reciprocals of Infinite Matrices

2.1 Reciprocals of Lower Semi-matrices, and Some Simple General Results 19

2.2 Some General Remarks on Reciprocals 21

2.3 The Bound of a Matrix 26

2.4 Two General Theorems on Reciprocals 29

2.5 A Theorem of Pólya 31

Examples 2 36

Chapter 3 Linear Equations in Infinite Matrices

3.1 Introduction 39

3.2 The Equations AX = B, XA = B 39

3.3 Transformation of an Infinite Matrix into a Diagonal Matrix 40

3.4 The Equation AX-XB = C, and the "Quantization" Equation AX - XA = I 47

3.5 An "Algebra" Theorem on AX - XA = I 51

Examples 3 54

Chapter 4 Divergent Sequences and Series

4.1 The Fundamental Theorems op Kojima-Schur and Silverman-Toeplitz 58

4.2 Analogues for Series ; β and γ-matrices 65

4.3 Examples of T and γ-matrices 68

4.4 Some Properties of T and γ-matrices Theorem of Steinhaus 73

4.5 Some Theorems of Agnew 79

4.6 Some General Properties of K, T, β, and γ-matrices 82

4.7 A Theorem on Bounded Divergent Sequences 88

4.8 A Theorem of Kaczmarz on Orthogonal Series 89

Examples 4 93

Chapter 5 Consistency, Mutual-Consistency, and Absolute Equivalence

5.1 Definitions 96

5.2 Consistency and Commutability 97

5.3 Commutability of Infinite Matrices 100

5.4 Absolute Equivalence for Bounded Sequences 105

5.5 Absolute Equivalence for Unbounded Sequences 109

5.6 Regular (Translative), and Absolutely Regular (Absolutely Translative), T-Limits 113

5.7 Comparison of Products of Different Methods of Summability 121

5.8 Note on a Memoir by Brudno 130

Examples 5 131

Chapter 6 The Core of a Sequence

6.1 Knopp's Core Theorem 137

6.2 Some Theorems of Agnew on the Core 141

6.3 Definitely Divergent Sequences 143

6.4 The Extended Core Theorem for Bounded Sequences 148

6.5 Two Theorems of A. Robinson on Absolute Equivalence and the Core 152

6.6 The Theorem of Steinhaus and the Core 157

Examples 6 169

Chapter 7 The Inefficiency Problem for Infinite Matrices

7.1 Inefficient Matrices with Left-hand K_r-Reciprocals 163

7.2 Inefficient Matrices having a Left-hand Reciprocal which is not a K_r-Matrix 165

7.3 Some Simple Results involving Absolute Equivalence 168

7.4 Matrices Efficient at a Point, or Isolated Set op Points, for Taylor Series outside the Circle of Convergence 169

7.5 Some Further Results on Inefficiency 173

7.6 On K-reciprocals of T-matrices 175

Examples 7 177

Chapter 8 The Efficiency Problem for Infinite Matrices

8.1 The Nature of the Efficiency Problem 181

8.2 Matrices Efficient for all Taylor Series in the Principal Star-domain 183

8.3 Matrices Efficient for all Taylor Series in Partial Star-domains 187

8.4 Lower Semi-matrices Efficient in Regions outside the Circle of Convergence 197

8.5 Efficiency for Bounded Sequences 200

8.6 On Summability of Sequences of O's and 1's 207

8.7 The "Right" Value for the Generalized Limit of a Bounded Divergent Sequence 213

Examples 8 220

Chapter 9 Hilbert Vector Space and Hilbert Matrices

9.1 Definitions 224

9.2 Strong and Weak Convergence 226

9.3 Vector Manifolds; Separability of Hubert Vector Space and its Consequences 230

9.4 The Bilinear Form 242

9.5 Bounds of H-matrices 254

9.6 Convolution (Faltung) Theorems; Reciprocals of H-matrices 262

9.7 Continuity in the Hilbert Space σ₂ 267

Examples 9 269

Chapter 10 Projective and Distance Convergence and Limit in Sequence Spaces

10.1 Some Different Types of Sequence Spaces 272

10.2 Coordinate Convergence and Projective Convergence 282

10.3 Projective Limit 287

10.4 Projective-bounded Sets 293

10.5 Strong Projective Convergence and Limit 302

10.6 Closure under Strong Projective Convergence 307

10.7 Some Properties of p-cgt Sequences 312

10.8 Distance Convergence and Limit 315

10.9 Baire's Category Theorem and the Banach-Steinhaus Theorem 317

Examples 10 321

Appendix 324

Bibliography of Memoirs and Books Referred to in the Text 327

Index of Names Referred to in the Text 341

General Index 343