Advanced Calculus An Introduction to Linear Analysis

by Richardson, Leonard F.

• ISBN13: 9780470232880
• ISBN10: 0470232889
• Edition: 1st
• Format: Hardcover
• Copyright: 2008-04-25
• Publisher: Wiley-Interscience

Summary

Features an introduction to advanced calculus and highlights its inherent concepts from linear algebraAdvanced Calculus reflects the unifying role of linear algebra in an effort to smooth readers' transition to advanced mathematics. The book fosters the development of complete theorem-proving skills through abundant exercises while also promoting a sound approach to the study. The traditional theorems of elementary differential and integral calculus are rigorously established, presenting the foundations of calculus in a way that reorients thinking toward modern analysis.Following an introduction dedicated to writing proofs, the book is divided into three parts:Part One explores foundational one-variable calculus topics from the viewpoint of linear spaces, norms, completeness, and linear functionals.Part Two covers Fourier series and Stieltjes integration, which are advanced one-variable topics.Part Three is dedicated to multivariable advanced calculus, including inverse and implicit function theorems and Jacobian theorems for multiple integrals.Numerous exercises guide readers through the creation of their own proofs, and they also put newly learned methods into practice. In addition, a "Test Yourself" section at the end of each chapter consists of short questions that reinforce the understanding of basic concepts and theorems. The answers to these questions and other selected exercises can be found at the end of the book along with an appendix that outlines key terms and symbols from set theory.Guiding readers from the study of the topology of the real line to the beginning theorems and concepts of graduate analysis, Advanced Calculus is an ideal text for courses in advanced calculus and introductory analysis at the upper-undergraduate and beginning-graduate levels. It also serves as a valuable reference for engineers, scientists, and mathematicians.

Author Biography

Leonard F. Richardson, PhD, is Herbert Huey McElveen Professor and Assistant Chair of the Department of Mathematics at Louisiana State University, where he is also Director of Graduate Studies in Mathematics. Dr. Richardson's research interests include harmonic analysis and representation theory.

Table of Contents

Preface	p. xiii
Acknowledgments	p. xix
Introduction	p. xxi
Advanced Calculus in One Variable
Real Numbers and Limits of Sequences	p. 3
The Real Number System	p. 3
Exercises	p. 7
Limits of Sequences & Cauchy Sequences	p. 8
Exercises	p. 12
The Completeness Axiom and Some Consequences	p. 13
Exercises	p. 18
Algebraic Combinations of Sequences	p. 19
Exercises	p. 21
The Bolzano-Weierstrass Theorem	p. 22
Exercises	p. 24
The Nested Intervals Theorem	p. 24
Exercises	p. 26
The Heine-Borel Covering Theorem	p. 27
Exercises	p. 30
Countability of the Rational Numbers	p. 31
Exercises	p. 35
Test Yourself	p. 37
Exercises	p. 37
Continuous Functions	p. 39
Limits of Functions	p. 39
Exercises	p. 43
Continuous Functions	p. 46
Exercises	p. 49
Some Properties of Continuous Functions	p. 50
Exercises	p. 53
Extreme Value Theorem and Its Consequences	p. 55
Exercises	p. 60
The Banach Space C[a, b]	p. 61
Exercises	p. 66
Test Yourself	p. 67
Exercises	p. 67
Riemann Integral	p. 69
Definition and Basic Properties	p. 69
Exercises	p. 74
The Darboux Integrability Criterion	p. 76
Exercises	p. 81
Integrals of Uniform Limits	p. 83
Exercises	p. 87
The Cauchy-Schwarz Inequality	p. 90
Exercises	p. 93
Test Yourself	p. 95
Exercises	p. 95
The Derivative	p. 99
Derivatives and Differentials	p. 99
Exercises	p. 103
The Mean Value Theorem	p. 105
Exercises	p. 109
The Fundamental Theorem of Calculus	p. 110
Exercises	p. 112
Uniform Convergence and the Derivative	p. 114
Exercises	p. 116
Cauchy's Generalized Mean Value Theorem	p. 117
Exercises	p. 121
Taylor's Theorem	p. 122
Exercises	p. 125
Test Yourself	p. 126
Exercises	p. 126
Infinite Series	p. 127
Series of Constants	p. 127
Exercises	p. 132
Convergence Tests for Positive Term Series	p. 134
Exercises	p. 137
Absolute Convergence and Products of Series	p. 138
Exercises	p. 146
The Banach Space l[subscript 1] and Its Dual Space	p. 148
Exercises	p. 153
Series of Functions: The Weierstrass M-Test	p. 154
Exercises	p. 157
Power Series	p. 158
Exercises	p. 161
Real Analytic Functions and C[superscript infinity] Functions	p. 162
Exercises	p. 167
Weierstrass Approximation Theorem	p. 169
Exercises	p. 173
Test Yourself	p. 174
Exercises	p. 174
Advanced Topics in one Variable
Fourier Series	p. 179
The Vibrating String and Trigonometric Series	p. 180
Exercises	p. 183
Euler's Formula and the Fourier Transform	p. 184
Exercises	p. 190
Bessel's Inequality and l[subscript 2]	p. 192
Exercises	p. 196
Uniform Convergence & Riemann Localization	p. 197
Exercises	p. 204
L[superscript 2]-Convergence & the Dual of l[superscript 2]	p. 205
Exercises	p. 208
Test Yourself	p. 212
Exercises	p. 212
The Riemann-Stieltjes Integral	p. 215
Functions of Bounded Variation	p. 216
Exercises	p. 220
Riemann-Stieltjes Sums and Integrals	p. 223
Exercises	p. 227
Riemann-Stieltjes Integrability Theorems	p. 228
Exercises	p. 230
The Riesz Representation Theorem	p. 231
Exercises	p. 239
Test Yourself	p. 241
Exercises	p. 241
Advanced Calculus in Several Variables
Euclidean Space	p. 245
Euclidean Space as a Complete Normed Vector Space	p. 245
Exercises	p. 249
Open Sets and Closed Sets	p. 252
Exercises	p. 254
Compact Sets	p. 256
Exercises	p. 258
Connected Sets	p. 259
Exercises	p. 261
Test Yourself	p. 263
Exercises	p. 263
Continuous Functions on Euclidean Space	p. 265
Limits of Functions	p. 265
Exercises	p. 268
Continuous Functions	p. 270
Exercises	p. 272
Continuous Image of a Compact Set	p. 274
Exercises	p. 276
Continuous Image of a Connected Set	p. 278
Exercises	p. 279
Test Yourself	p. 280
Exercises	p. 280
The Derivative in Euclidean Space	p. 283
Linear Transformations and Norms	p. 283
Exercises	p. 286
Differentiable Functions	p. 289
Exercises	p. 295
The Chain Rule in Euclidean Space	p. 298
The Mean Value Theorem	p. 300
Taylor's Theorem	p. 301
Exercises	p. 303
Inverse Functions	p. 305
Exercises	p. 309
Implicit Functions	p. 311
Exercises	p. 317
Tangent Spaces and Lagrange Multipliers	p. 322
Exercises	p. 327
Test Yourself	p. 328
Exercises	p. 328
Riemann Integration in Euclidean Space	p. 331
Definition of the Integral	p. 331
Exercises	p. 336
Lebesgue Null Sets and Jordan Null Sets	p. 338
Exercises	p. 341
Lebesgue's Criterion for Riemann Integrability	p. 342
Exercises	p. 344
Fubini's Theorem	p. 346
Exercises	p. 349
Jacobian Theorem for Change of Variables	p. 351
Exercises	p. 355
Test Yourself	p. 357
Exercises	p. 357
Set Theory	p. 359
Terminology and Symbols	p. 359
Exercises	p. 363
Paradoxes	p. 363
Problem Solutions	p. 365
References	p. 379
Index	p. 381
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