A Concise Introduction
Buch, Englisch, 592 Seiten, Format (B × H): 161 mm x 240 mm, Gewicht: 1029 g
ISBN: 978-0-470-10796-6
Verlag: Wiley
A self-contained introduction to the fundamentals of mathematical analysis
Mathematical Analysis: A Concise Introduction presents the foundations of analysis and illustrates its role in mathematics. By focusing on the essentials, reinforcing learning through exercises, and featuring a unique "learn by doing" approach, the book develops the reader's proof writing skills and establishes fundamental comprehension of analysis that is essential for further exploration of pure and applied mathematics. This book is directly applicable to areas such as differential equations, probability theory, numerical analysis, differential geometry, and functional analysis.
Mathematical Analysis is composed of three parts:
?Part One presents the analysis of functions of one variable, including sequences, continuity, differentiation, Riemann integration, series, and the Lebesgue integral. A detailed explanation of proof writing is provided with specific attention devoted to standard proof techniques. To facilitate an efficient transition to more abstract settings, the results for single variable functions are proved using methods that translate to metric spaces.
?Part Two explores the more abstract counterparts of the concepts outlined earlier in the text. The reader is introduced to the fundamental spaces of analysis, including Lp spaces, and the book successfully details how appropriate definitions of integration, continuity, and differentiation lead to a powerful and widely applicable foundation for further study of applied mathematics. The interrelation between measure theory, topology, and differentiation is then examined in the proof of the Multidimensional Substitution Formula. Further areas of coverage in this section include manifolds, Stokes' Theorem, Hilbert spaces, the convergence of Fourier series, and Riesz' Representation Theorem.
?Part Three provides an overview of the motivations for analysis as well as its applications in various subjects. A special focus on ordinary and partial differential equations presents some theoretical and practical challenges that exist in these areas. Topical coverage includes Navier-Stokes equations and the finite element method.
Mathematical Analysis: A Concise Introduction includes an extensive index and over 900 exercises ranging in level of difficulty, from conceptual questions and adaptations of proofs to proofs with and without hints. These opportunities for reinforcement, along with the overall concise and well-organized treatment of analysis, make this book essential for readers in upper-undergraduate or beginning graduate mathematics courses who would like to build a solid foundation in analysis for further work in all analysis-based branches of mathematics.
Autoren/Hrsg.
Weitere Infos & Material
Preface xi
Part I: Analysis of Functions of a Single Real Variable
1 The Real Numbers 1
1.1 Field Axioms 1
1.2 Order Axioms 4
1.3 Lowest Upper and Greatest Lower Bounds 8
1.4 Natural Numbers, Integers, and Rational Numbers 11
1.5 Recursion, Induction, Summations, and Products 17
2 Sequences of Real Number V 25
2.1 Limits 25
2.2 Limit Laws 30
2.3 Cauchy Sequences 36
2.4 Bounded Sequences 40
2.5 Infinite Limits 44
3 Continuous Functions 49
3.1 Limits of Functions 49
3.2. Limit Laws 52
3.3 One-Sided Limits and Infinite Limits 56
3.4 Continuity 59
3.5 Properties of Continuous Functions 66
3.6 Limits at Infinity 69
4 Differentiable Functions 71
4.1 Differentiability 71
4.2 Differentiation Rules 74
4.3 Rolle's Theorem and the Mean Value Theorem 80
5 The Riemann Integral I 85
5.1 Riemann Sums and the Integral 85
5.2 Uniform Continuity and Integrability of Continuous Functions 91
5.3 The Fundamental Theorem of Calculus 95
5.4 The Darboux Integral 97
6 Series of Real Numbers I 101
6.1 Series as a Vehicle to Define Infinite Sums 101
6.2 Absolute Convergence and Unconditional Convergence 108
7 Some Set Theory 117
7.1 The Algebra of Sets 117
7.2 Countable Sets 122
7.3 Uncountable Sets 124
8 The Riemann Integral II 127
8.1 Outer Lebesgue Measure 127
8.2 Lebesgue's Criterion for Riemann Integrability 131
8.3 More Integral Theorems 136
8.4 Improper Riemann Integrals 140
9 The Lebesgue Integral 145
9.1 Lebesgue Measurable Sets 147
9.2 Lebesgue Measurable Functions 153
9.3 Lebesgue Integration 158
9.4 Lebesgue Integrals versus Riemann Integrals 165
10 Series of Real Numbers II 169
10.1 Limits Superior and Inferior 169
