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Real Analysis
Frank Morgan, Williams College, Williamstown, MA
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2005; 151 pp; hardcover
ISBN-10: 0-8218-3670-6
ISBN-13: 978-0-8218-3670-5
List Price: US$43
Member Price: US$34.40
Order Code: REAL
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This book is written by award-winning author, Frank Morgan. It offers a simple and sophisticated point of view, reflecting Morgan's insightful teaching, lecturing, and writing style.

Intended for undergraduates studying real analysis, this book builds the theory behind calculus directly from the basic concepts of real numbers, limits, and open and closed sets in \(\mathbb{R}^n\). It gives the three characterizations of continuity: via epsilon-delta, sequences, and open sets. It gives the three characterizations of compactness: as "closed and bounded," via sequences, and via open covers. Topics include Fourier series, the Gamma function, metric spaces, and Ascoli's Theorem.

This concise text not only provides efficient proofs, but also shows students how to derive them. The excellent exercises are accompanied by select solutions. Ideally suited as an undergraduate textbook, this complete book on real analysis will fit comfortably into one semester.

Frank Morgan received the first Haimo Award for distinguished college teaching from the Mathematical Association of America. He has also garnered top teaching awards from Rice University (Houston, TX) and MIT (Cambridge, MA).

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Readership

Undergraduate students interested in real analysis.

Reviews

"Reading your book is a refreshingly delightful change from the usual emphasis on series, rather than topology, as a foundation of analysis."

-- Robert Jones, University of Dusseldorf

Table of Contents

Part I: Real numbers and limits
  • Numbers and logic
  • Infinity
  • Sequences
  • Functions and limits
Part II: Topology
  • Open and closed sets
  • Continuous functions
  • Composition of functions
  • Subsequences
  • Compactness
  • Existence of maximum
  • Uniform continuity
  • Connected sets and the intermediate value theorem
  • The Cantor set and fractals
Part III: Calculus
  • The derivative and the mean value theorem
  • The Riemann integral
  • The fundamental theorem of calculus
  • Sequences of functions
  • The Lebesgue theory
  • Infinite series \(\sum a_n\)
  • Absolute convergence
  • Power series
  • Fourier series
  • Strings and springs
  • Convergence of Fourier series
  • The exponential function
  • Volumes of \(n\)-balls and the gamma function
Part IV: Metric spaces
  • Metric spaces
  • Analysis on metric spaces
  • Compactness in metric spaces
  • Ascoli's theorem
  • Partial solutions to exercises
  • Greek letters
  • Index
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