Real Mathematical Analysis

ISBN-13: 9781441929419 / Angielski / Miękka / 2010 / 440 str.

Charles C. Pugh

Opis

Was plane geometry your favorite math course in high school? Did you like proving theorems? Are you sick of memorizing integrals? If so, real analysis could be your cup of tea. In contrast to calculus and elementary algebra, it involves neither formula manipulation nor applications to other fields of science. None. It is pure mathematics, and I hope it appeals to you, the budding pure mathematician. Berkeley, California, USA CHARLES CHAPMAN PUGH 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 x Contents 3 Functions of a Real Variable 139 1 Differentiation. . . . 139 2 Riemann Integration 154 Series . . 179 3 Exercises 186 4 Function Spaces 201 1 Uniform Convergence and CO a, b] 201 2 Power Series . . . . . . . . . . . . 211 3 Compactness and Equicontinuity in CO . 213 4 Uniform Approximation in CO 217 Contractions and ODE's . . . . . . . . 228 5 6* Analytic Functions . . . . . . . . . . . 235 7* Nowhere Differentiable Continuous Functions . 240 8* Spaces of Unbounded Functions 248 Exercises . . . . . 251 267 5 Multivariable Calculus 1 Linear Algebra . . 267 2 Derivatives. . . . 271 3 Higher derivatives . 279 4 Smoothness Classes . 284 5 Implicit and Inverse Functions 286 290 6* The Rank Theorem 296 7* Lagrange Multipliers 8 Multiple Integrals . .

Recenzje

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)

Spis treści

Real Numbers * A Taste of Topology * Functions of a Real Variable * Function Spaces * Multivariable Calculus * Lebesgue Theory * Index

Blurb

Was plane geometry your favorite math course in high school? Did you like proving theorems? Are you sick of memorizing integrals? If so, real analysis could be your cup of tea. In contrast to calculus and elementary algebra, it involves neither formula manipulation nor applications to other fields of science. None. It is Pure Mathematics, and it is sure to appeal to the budding pure mathematician.
In this new introduction to undergraduate real analysis the author takes a different approach from past presentations of the subject, 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 such as Dieudonne, Littlewood and Osserman. This book is based on the honors version of a course which the author has taught many times over the last 35 years at Berkeley. The book contains an excellent selection of more than 500 exercises.