Mathematical Analysis A Concise Introduction

by Schrö; der, Bernd S. W.

• ISBN13: 9780470107966
• ISBN10: 0470107960
• Edition: 1st
• Format: Hardcover
• Copyright: 2007-11-12
• Publisher: Wiley-Interscience

Summary

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.

Author Biography

Bernd S.W. Schroder, PhD, is Edmondson/Crump Professor in the Program of Mathematics and Statistics at Louisiana Tech University. Dr. Schröder is the author of over thirty refereed journal articles on subjects such as ordered sets, probability theory, graph theory, harmonic analysis, computer science, and education. He earned his PhD in mathematics from Kansas State University in 1992.

Table of Contents

Table of Contents	p. v
Preface	p. xi
Analysis of Functions of a Single Real Variable
The Real Numbers	p. 1
Field Axioms	p. 1
Order Axioms	p. 4
Lowest Upper and Greatest Lower Bounds	p. 8
Natural Numbers, Integers, and Rational Numbers	p. 11
Recursion, Induction, Summations, and Products	p. 17
Sequences of Real Numbers	p. 25
Limits	p. 25
Limit Laws	p. 30
Cauchy Sequences	p. 36
Bounded Sequences	p. 40
Infinite Limits	p. 44
Continuous Functions	p. 49
Limits of Functions	p. 49
Limit Laws	p. 52
One-Sided Limits and Infinite Limits	p. 56
Continuity	p. 59
Properties of Continuous Functions	p. 66
Limits at Infinity	p. 69
Differentiable Functions	p. 71
Differentiability	p. 71
Differentiation Rules	p. 74
Rolle's Theorem and the Mean Value Theorem	p. 80
The Riemann Integral I	p. 85
Riemann Sums and the Integral	p. 85
Uniform Continuity and Integrability of Continuous Functions	p. 91
The Fundamental Theorem of Calculus	p. 95
The Darboux Integral	p. 97
Series of Real Numbers I	p. 101
Series as a Vehicle To Define Infinite Sums	p. 101
Absolute Convergence and Unconditional Convergence	p. 108
Some Set Theory	p. 117
The Algebra of Sets	p. 117
Countable Sets	p. 122
Uncountable Sets	p. 124
The Riemann Integral II	p. 127
Outer Lebesgue Measure	p. 127
Lebesgue's Criterion for Riemann Integrability	p. 131
More Integral Theorems	p. 136
Improper Riemann Integrals	p. 140
The Lebesgue Integral	p. 145
Lebesgue Measurable Sets	p. 147
Lebesgue Measurable Functions	p. 153
Lebesgue Integration	p. 158
Lebesgue Integrals versus Riemann Integrals	p. 165
Series of Real Numbers II	p. 169
Limits Superior and Inferior	p. 169
The Root Test and the Ratio Test	p. 172
Power Series	p. 175
Sequences of Functions	p. 179
Notions of Convergence	p. 179
Uniform Convergence	p. 182
Transcendental Functions	p. 189
The Exponential Function	p. 189
Sine and Cosine	p. 193
L'Hopital's Rule	p. 199
Numerical Methods	p. 203
Approximation with Taylor Polynomials	p. 204
Newton's Method	p. 208
Numerical Integration	p. 214
Analysis in Abstract Spaces
Integration on Measure Spaces	p. 225
Measure Spaces	p. 225
Outer Measures	p. 230
Measurable Functions	p. 234
Integration of Measurable Functions	p. 235
Monotone and Dominated Convergence	p. 238
Convergence in Mean, in Measure, and Almost Everywhere	p. 242
Product [sigma]-Algebras	p. 245
Product Measures and Fubini's Theorem	p. 251
The Abstract Venues for Analysis	p. 255
Abstraction I: Vector Spaces	p. 255
Representation of Elements: Bases and Dimension	p. 259
Identification of Spaces: Isomorphism	p. 262
Abstraction II: Inner Product Spaces	p. 264
Nicer Representations: Orthonormal Sets	p. 267
Abstraction III: Normed Spaces	p. 269
Abstraction IV: Metric Spaces	p. 275
L[superscript p] Spaces	p. 278
Another Number Field: Complex Numbers	p. 281
The Topology of Metric Spaces	p. 287
Convergence of Sequences	p. 287
Completeness	p. 291
Continuous Functions	p. 296
Open and Closed Sets	p. 301
Compactness	p. 309
The Normed Topology of R[superscript d]	p. 316
Dense Subspaces	p. 322
Connectedness	p. 330
Locally Compact Spaces	p. 333
Differentiation in Normed Spaces	p. 341
Continuous Linear Functions	p. 342
Matrix Representation of Linear Functions	p. 348
Differentiability	p. 353
The Mean Value Theorem	p. 360
How Partial Derivatives Fit In	p. 362
Multilinear Functions (Tensors)	p. 369
Higher Derivatives	p. 373
The Implicit Function Theorem	p. 380
Measure, Topology, and Differentiation	p. 385
Lebesgue Measurable Sets in R[superscript d]	p. 385
C[infinity] and Approximation of Integrable Functions	p. 391
Tensor Algebra and Determinants	p. 397
Multidimensional Substitution	p. 407
Introduction to Differential Geometry	p. 421
Manifolds	p. 421
Tangent Spaces and Differentiable Functions	p. 427
Differential Forms, Integrals Over the Unit Cube	p. 434
k-Forms and Integrals Over k-Chains	p. 443
Integration on Manifolds	p. 452
Stokes' Theorem	p. 458
Hilbert Spaces	p. 463
Orthonormal Bases	p. 463
Fourier Series	p. 467
The Riesz Representation Theorem	p. 475
Applied Analysis
Physics Background	p. 483
Harmonic Oscillators	p. 484
Heat and Diffusion	p. 486
Separation of Variables, Fourier Series, and Ordinary Differential Equations	p. 490
Maxwell's Equations	p. 493
The Navier Stokes Equation for the Conservation of Mass	p. 496
Ordinary Differential Equations	p. 505
Banach Space Valued Differential Equations	p. 505
An Existence and Uniqueness Theorem	p. 508
Linear Differential Equations	p. 510
The Finite Element Method	p. 513
Ritz-Galerkin Approximation	p. 513
Weakly Differentiable Functions	p. 518
Sobolev Spaces	p. 524
Elliptic Differential Operators	p. 532
Finite Elements	p. 536
Conclusion and Outlook	p. 544
Appendices
Logic	p. 545
Statements	p. 545
Negations	p. 546
Set Theory	p. 547
The Zermelo-Fraenkel Axioms	p. 547
Relations and Functions	p. 548
Natural Numbers, Integers, and Rational Numbers	p. 549
The Natural Numbers	p. 549
The Integers	p. 550
The Rational Numbers	p. 550
Bibliography	p. 551
Index	p. 553
Table of Contents provided by Ingram. All Rights Reserved.