Real Analysis

1. Front Matter

   Pages i-xxxii

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2. Preliminaries

   • Emmanuele DiBenedetto

   Pages 1-15

3. Topologies and Metric Spaces

   • Emmanuele DiBenedetto

   Pages 17-66

4. Measuring Sets

   • Emmanuele DiBenedetto

   Pages 67-131

5. The Lebesgue Integral

   • Emmanuele DiBenedetto

   Pages 133-191

6. Topics on Measurable Functions of Real Variables

   • Emmanuele DiBenedetto

   Pages 193-245

7. The \(L^p\) Spaces

   • Emmanuele DiBenedetto

   Pages 247-312

8. Banach Spaces

   • Emmanuele DiBenedetto

   Pages 313-377

9. Spaces of Continuous Functions, Distributions, and Weak Derivatives

   • Emmanuele DiBenedetto

   Pages 379-429

10. Topics on Integrable Functions of Real Variables

    • Emmanuele DiBenedetto

    Pages 431-489

11. Embeddings of \(W^{1,p}(E)\) into \(L^q(E)\)

    • Emmanuele DiBenedetto

    Pages 491-539

12. Topics on Weakly Differentiable Functions

    • Emmanuele DiBenedetto

    Pages 541-578

13. Back Matter

    Pages 579-596

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The second edition of this classic textbook presents a rigorous and self-contained introduction to real analysis with the goal of providing a solid foundation for future coursework and research in applied mathematics.  Written in a clear and concise style, it covers all of the necessary subjects as well as those often absent from standard introductory texts.  Each chapter features a “Problems and Complements” section that includes additional material that briefly expands on certain topics within the chapter and numerous exercises for practicing the key concepts.

The first eight chapters explore all of the basic topics for training in real analysis, beginning with a review of countable sets before moving on to detailed discussions of measure theory, Lebesgue integration, Banach spaces, functional analysis, and weakly differentiable functions.  More topical applications are discussed in the remaining chapters, such as maximal functions, functions of bounded mean oscillation, rearrangements, potential theory, and the theory of Sobolev functions.  This second edition has been completely revised and updated and contains a variety of new content and expanded coverage of key topics, such as new exercises on the calculus of distributions, a proof of the Riesz convolution, Steiner symmetrization, and embedding theorems for functions in Sobolev spaces.  

Ideal for either classroom use or self-study, Real Analysis is an excellent textbook both for students discovering real analysis for the first time and for mathematicians and researchers looking for a useful resource for reference or review.

Praise for the First Edition:

“[This book] will be extremely useful as a text.  There is certainly enough material for a year-long graduate course, but judicious selection would make it possible to use this most appealing book in a one-semester course for well-prepared students.”  

—Mathematical Reviews

• real analysis
• measure theory
• functional analysis
• Lebesgue integration
• weakly differentiable functions
• partial differential equations

“The book is a valuable, comprehensive reference source on real analysis. The first eight chapters cover core material that is part of most courses taught on the subject, followed by a collection of special topics that stay within the framework of real analysis. In addition to the content, what makes the book especially useful as a reference source is its organization. … Summing Up: Recommended. Graduate students and faculty. This work should be used solely as a reference.” (M. Bona, Choice, Vol. 54 (9), May, 2017)

“The reader can find many interesting details which serve to illuminate the diamonds of analysis. The list of references contains the main books and articles which form the modern real analysis. The book can be recommended as one of the main readings on real analysis for those who are interested in this subject and its numerous applications.” (Sergei V. Rogosin, zbMATH 1353.26001, 2017)

• Nashville, USA

  Emmanuele DiBenedetto

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