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
Part of the book series: Undergraduate Texts in Mathematics (UTM)
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Table of contents (6 chapters)
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Front Matter
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Back Matter
Keywords
About this book
Based on an honors course taught by the author at UC Berkeley, this introduction to undergraduate real analysis gives a different emphasis by stressing the importance of pictures and hard problems. Topics include: a natural construction of the real numbers, four-dimensional visualization, basic point-set topology, function spaces, multivariable calculus via differential forms (leading to a simple proof of the Brouwer Fixed Point Theorem), and a pictorial treatment of Lebesgue theory. Over 150 detailed illustrations elucidate abstract concepts and salient points in proofs. The exposition is informal and relaxed, with many helpful asides, examples, some jokes, and occasional comments from mathematicians, such as Littlewood, Dieudonné, and Osserman. This book thus succeeds in being more comprehensive, more comprehensible, and more enjoyable, than standard introductions to analysis.
New to the second edition of Real Mathematical Analysis is a presentation of Lebesgue integration done almost entirely using the undergraph approach of Burkill. Payoffs include: concise picture proofs of the Monotone and Dominated Convergence Theorems, a one-line/one-picture proof of Fubini's theorem from Cavalieri’s Principle, and, in many cases, the ability to see an integral result from measure theory. The presentation includes Vitali’s Covering Lemma, density points — which are rarely treated in books at this level — and the almost everywhere differentiability of monotone functions. Several new exercises now join a collection of over 500 exercises that pose interesting challenges and introduce special topics to the student keen on mastering this beautiful subject.
Authors and Affiliations
About the author
Bibliographic Information
Book Title: Real Mathematical Analysis
Authors: Charles C. Pugh
Series Title: Undergraduate Texts in Mathematics
DOI: https://doi.org/10.1007/978-3-319-17771-7
Publisher: Springer Cham
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: Springer Nature Switzerland AG 2015
Hardcover ISBN: 978-3-319-17770-0Published: 07 August 2015
Softcover ISBN: 978-3-319-33042-6Published: 15 October 2016
eBook ISBN: 978-3-319-17771-7Published: 29 July 2015
Series ISSN: 0172-6056
Series E-ISSN: 2197-5604
Edition Number: 2
Number of Pages: XI, 478
Number of Illustrations: 1 illustrations in colour
Topics: Measure and Integration, Real Functions, Sequences, Series, Summability