Real Mathematical Analysis

by Charles Chapman Pugh

Overview

Elucidates abstract concepts and salient points in proofs with over 150 detailed illustrations

Treats the rigorous foundations of both single and multivariable Calculus

Gives an intuitive presentation of Lebesgue integration using the undergraph approach of Burkill

Includes over 500 exercises that are interesting and thought-provoking, not merely routine

Product Details

ISBN-13:	9783319330426
Publisher:	Springer International Publishing
Publication date:	08/12/2016
Series:	Undergraduate Texts in Mathematics
Edition description:	Softcover reprint of the original 2nd ed. 2015
Pages:	478
Product dimensions:	7.01(w) x 10.00(h) x 0.04(d)

About the Author

Charles C. Pugh is Professor Emeritus at the University of California, Berkeley. His research interests include geometry and topology, dynamical systems, and normal hyperbolicity.

Table of Contents

1	Real Numbers	1
1	Preliminaries	1
2	Cuts	10
3	Euclidean Space	21
4	Cardinality	28
5	Comparing Cardinalities	34
6	The Skeleton of Calculus	36
	Exercises	40
2	A Taste of Topology	51
1	Metric Space Concepts	51
2	Compactness	76
3	Connectedness	82
4	Coverings	88
5	Cantor Sets	95
6	Cantor Set Lore	99
7	Completion	108
	Exercises	115
3	Functions of a Real Variable	139
1	Differentiation	139
2	Riemann Integration	154
3	Series	179
	Exercises	186
4	Function Spaces	201
1	Uniform Convergence and C[superscript 0 a,b]	201
2	Power Series	211
3	Compactness and Equicontinuity in C[superscript 0]	213
4	Uniform Approximation in C[superscript 0]	217
5	Contractions and ODE's	228
6	Analytic Functions	235
7	Nowhere Differentiable Continuous Functions	240
8	Spaces of Unbounded Functions	248
	Exercises	251
5	Multivariable Calculus	267
1	Linear Algebra	267
2	Derivatives	271
3	Higher derivatives	279
4	Smoothness Classes	284
5	Implicit and Inverse Functions	286
6	The Rank Theorem	290
7	Lagrange Multipliers	296
8	Multiple Integrals	300
9	Differential Forms	313
10	The General Stokes' Formula	325
11	The Brouwer Fixed Point Theorem	334
Appendix A	Perorations of Dieudonne	337
Appendix B	The History of Cavalieri's Principle	338
Appendix C	A Short Excursion into the Complex Field	339
Appendix D	Polar Form	340
Appendix E	Determinants	342
	Exercises	345
6	Lebesgue Theory	363
1	Outer measure	363
2	Measurability	367
3	Regularity	371
4	Lebesgue integrals	376
5	Lebesgue integrals as limits	383
6	Italian Measure Theory	387
7	Vitali coverings and density points	391
8	Lebesgue's Fundamental Theorem of Calculus	396
9	Lebesgue's Last Theorem	401
Appendix A	Translations and Nonmeasurable sets	407
Appendix B	The Banach-Tarski Paradox	409
Appendix C	Riemann integrals as undergraphs	409
Appendix D	Littlewood's Three Principles	411
Appendix E	Roundness	412
Appendix F	Money	413
	Suggested Reading	414
	Bibliography	415
	Exercises	417
	Index	431

Editorial Reviews

s. I should add here that Pugh succeeds in transferring his love and enthusiasm for this material to the reader. … As a mathematics student, I would have loved to have this as a textbook to be taught my first analysis course. As a teacher … I would love to use it as lecture notes." (Adhemar Bultheel, Belgian Mathematical Society - Simon Stevin Bulletin, Vol. 11 (1), 2004)

"The author of this undergraduate text believes that real analysis is the jewel in the crown of pure mathematics. … This text is based upon many years of teaching the analysis course at Berkeley. The exposition is chatty and easygoing, while managing to cover all of the basic ideas carefully and thoroughly. … The text is complemented by an excellent index and frequent suggestions for further reading. I can recommend this book to serious undergraduates who want to get into real analysis … ." (Gerry Leversha, The Mathematical Gazette, Vol. 88 (551), 2004)

"This book is a new introduction to undergraduate real analysis. … The exposition is informal and relaxed, with an excellent selection of more than 500 exercises. The occasional comments from mathematicians … make the text really enjoyable. … To sum up, this undergraduate … textbook contains a wealth of information. It is written in a concise, but always clear and well-readable way. … It should have a great appeal to the students of (under) graduate courses as well as to budding pure mathematicians." (Ferenc Móricz, Acta Scientiarum Mathematicarum, Vol. 69, 2003)

"Pugh’s book … is not a typical book. … it very successfully (and atypically) manages to convey the look and feel of an engaging classroom lecture while maintaining the highest level of rigor and care. … This makes this well-crafted book very appealing as a resource for an honors section, but it also should be in any undergraduate library as a source of ideas and supplementary problems for faculty or as a challenge for strong students. An excellent book in an excellent series. Highly recommended." (J. Feroe, CHOICE, September, 2002)

"This book is suited for a two-semester course in real analysis for upper-level undergraduate students who major in mathematics. … The book is very well written. The style is lively and engaging. Intuition is stimulated and metaphors are used throughout the book, without compromising rigor. … The exercises are numerous and they vary from straightforward to very challenging … . This is a book for the highly motivated student. He/she will get from this book a good grasp of analysis: concepts and techniques." (Sherif T. El-Helaly, Mathematical Reviews, 2003 e)

"The book under review is an introduction to the basics of real analysis. … A special feature of the exposition is its emphasis on the explanation of mathematical concepts by figures … . The book can be used for self-study. … The book can be highly recommended as an introduction to real analysis." (Joachim Naumann, Zentralblatt MATH, Vol. 1003 (3), 2003)

"In this new introduction to undergraduate real analysis, the author takes a different approach … by stressing the importance of pictures in mathematics and hard problems. The exposition is informal and relaxed, with many helpful asides, examples, and occasional comments from mathematicians … . This book is based on the honors version of a course that the author has taught many times, over the last thirty-five years, at the University of California, Berkeley." (L’ Enseignement Mathematique, Issue 1-2, 2002)

From the Publisher