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  • © 2015

Real Mathematical Analysis

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  • 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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  • ISBN: 978-3-319-17771-7
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Table of contents (6 chapters)

  1. Front Matter

    Pages i-xi
  2. Real Numbers

    • Charles C. Pugh
    Pages 1-55
  3. A Taste of Topology

    • Charles C. Pugh
    Pages 57-148
  4. Functions of a Real Variable

    • Charles C. Pugh
    Pages 149-210
  5. Function Spaces

    • Charles C. Pugh
    Pages 211-276
  6. Multivariable Calculus

    • Charles C. Pugh
    Pages 277-382
  7. Lebesgue Theory

    • Charles C. Pugh
    Pages 383-465
  8. Back Matter

    Pages 467-478

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.

Keywords

  • Brouwer fixed point theorem
  • Lebesgue integral
  • Riemann integral
  • calculus
  • mathematical analysis
  • multivariable calculus
  • nowhere differentiable continuous function
  • point-set topology
  • real analysis
  • real numbers
  • uniform convergence

Authors and Affiliations

  • Department of Mathematics, University of California, Berkeley, Berkeley, USA

    Charles C. Pugh

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.

Bibliographic Information

Buying options

eBook
USD 44.99
Price excludes VAT (USA)
  • ISBN: 978-3-319-17771-7
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book
USD 59.99
Price excludes VAT (USA)
Hardcover Book
USD 59.99
Price excludes VAT (USA)