Korevaar Mathematical Methods

Linear Algebra / Normed Spaces / Distributions / Integration
1. Auflage 2014
ISBN: 978-1-4832-7074-6
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 518 Seiten, Web PDF

Mathematical Methods, Volume I: Linear Algebra, Normed Spaces, Distributions, Integration focuses on advanced mathematical tools used in applications and the basic concepts of algebra, normed spaces, integration, and distributions. The publication first offers information on algebraic theory of vector spaces and introduction to functional analysis. Discussions focus on linear transformations and functionals, rectangular matrices, systems of linear equations, eigenvalue problems, use of eigenvectors and generalized eigenvectors in the representation of linear operators, metric and normed vector spaces, and delta sequences and convergence and approximation. The text then examines the Lebesgue integral, including approximation of integrable functions and applications, integration of sequences and series, functions of bounded variation and the Stieltjes integral, and multiple integrals. Curves and integrals, holomorphic functions and integrals in the complex plane, and multiple integrals are also discussed. The book is a valuable reference for students in the physical sciences, mathematics students interested in applications, and mathematically oriented engineering students.

Autoren/Hrsg.

Korevaar, Jacob

Inhaltsverzeichnis

1;Front Cover;1
2;Linear Algebra / Normed Spaces / Distributions / Integration;4
3;Copyright Page;5
4;Table of Contents;10
5;PREFACE;6
6;CHAPTER ONE. ALGEBRAIC THEORY
OF VECTOR SPACES;14
6.1;1. VECTOR SPACES;16
6.2;2. GENERATING SETS, LINEARLY INDEPENDENT SETS, AND BASES;25
6.3;3. DIMENSION OF A VECTOR SPACE;34
6.4;4. LINEAR TRANSFORMATIONS;43
6.5;5. LINEAR FUNCTIONALS;52
6.6;6. RECTANGULAR MATRICES;63
6.7;7. SQUARE MATRICES;77
6.8;8. DETERMINANTS;86
6.9;9. SYSTEMS OF LINEAR EQUATIONS;94
6.10;10. EIGENVALUE PROBLEMS;107
6.11;11. USE OF EIGENVECTORS AND GENERALIZED EIGENVECTORS IN THE REPRESENTATION OF LINEAR OPERATORS;121
6.12;12. FURTHER REPRESENTATION THEOREMS FOR LINEAR OPERATORS AND APPLICATIONS;134
6.13;13. ALGEBRAIC THEORY OF TENSORS;157
6.14;BIBLIOGRAPHY;170
7;CHAPTER TWO. INTRODUCTION TO FUNCTIONAL ANALYSIS. DISTRIBUTIONS;173
7.1;1. VARIOUS KINDS OF CONVERGENCE;175
7.2;2. METRIC SPACES;190
7.3;3. NORMED VECTOR SPACES;203
7.4;4. COMPLETENESS AND COMPLETION;219
7.5;5. THE BANACH SPACES L, LP, AND C;239
7.6;6. CONTINUOUS LINEAR FUNCTIONALS;262
7.7;7. DISTRIBUTIONS OR GENERALIZED FUNCTIONS (PART 1);276
7.8;8. DISTRIBUTIONS OR GENERALIZED FUNCTIONS (PART 2);301
7.9;9. DELTA SEQUENCES AND CONVERGENCE AND APPROXIMATION;331
7.10;10. SPANNING SETS AND SCHAUDER BASES;357
7.11;BIBLIOGRAPHY;366
8;CHAPTER THREE. THE LEBESGUE INTEGRAL AND RELATED TOPICS;369
8.1;1. DEFINITION OF THE LEBESGUE INTEGRAL;372
8.2;2. APPROXIMATION OF INTEGRABLE FUNCTIONS AND APPLICATIONS;386
8.3;3. INTEGRATION OF SEQUENCES AND SERIES;394
8.4;4. LEBESGUE MEASURE;408
8.5;5. FUNCTIONS OF BOUNDED VARIATION AND THE STIELTJES INTEGRAL;413
8.6;6. RULES BASED ON THE USE OF INDEFINITE INTEGRALS;426
8.7;7. MULTIPLE INTEGRALS;439
8.8;8. CURVES AND LINE INTEGRALS;463
8.9;9. HOLOMORPHIC FUNCTIONS AND INTEGRALS IN THE COMPLEX PLANE;478
8.10;BIBLIOGRAPHY;499
9;SUBJECT AND AUTHOR INDEX;502
10;INDEX OF NOTATION;516