Description: This final text in the Zakon Series on Mathematics Analysis follows the release of the author's Basic Concepts of Mathematics and the award-winning Mathematical Analysis I and completes the material on Real Analysis that is the foundation for later courses in functional analysis, harmonic analysis, probability theory, etc. The first chapter extends calculus to n-dimensional Euclidean space and, more generally, Banach spaces, covering the inverse function theorem, the implicit function theorem, Taylor expansions, etc. Some basic theorems in functional analysis, including the open mapping theorem and the Banach-Steinhaus uniform boundedness principle, are also proved. The text then moves to measure theory, with a complete discussion of outer measures, Lebesgue measure, Lebesgue-Stieltjes measures, and differentiation of set functions. The discussion of measurable functions and integration in the following chapter follows an innovative approach, carefully choosing one of the equivalent definitions of measurable functions that allows the most intuitive development of the material. Fubini's theorem, the Radon-Nikodym theorem, and the basic convergence theorems (Fatou's lemma, the monotone convergence theorem, dominated convergence theorem) are covered. Finally, a chapter relates antidifferentiation to Lebesgue theory, Cauchy integrals, and convergence of parametrized integrals. Nearly 500 exercises allow students to develop their skills in the area.

Audience: This text is appropriate for any second course in real analysis or mathematical analysis, whether at the undergraduate or graduate level.

This text covers similar material to the books (we give US list prices for current editions at Amazon on December 18, 2009): Real and Complex Analysis by W. Rudin ($93.75); Measure and Integral: An Introduction to Real Analysis by R. Wheeden and A. Zygmund ($62.86); Real Analysis: Theory of measure and Integration by J. Yeh ($69.00); Measure Theory by P. R. Halmos ($64.95); The Elements of Integration and Lebesgue Measure by R. G. Bartle ($76.19); etc.

Publishing Information: Mathematical Analysis II, by Elias Zakon, ISBN 978-1-931705-03-8, published by The Trillia Group, 2009. 424+xii pages, 483 exercises, 8 figures, hypertextual cross-references, hyperlinked index of terms. Download size: 2317 to 2630 KB, depending on format. Currrent version released May 11, 2017.

Excerpts from the Text: The Terms and Conditions, the Preface, the Table of Contents, and the Index.

Reviews: You can add your own review of this book at The Assayer.

Terms and Conditions: All uses of this text are subject to the Terms and Conditions contained in this text. As part of these terms, we license this text free of charge to students using it for self-study, and to lecturers evaluating it as a required or recommended text for a course. All other uses of this text are subject to a charge of $10US for individual use and $300US for use by all individuals at a single site of a college or university. Special note: Thanks to funding by the Saylor Foundation, Chapter 6 of Mathematical Analysis II has been licensed under a Creative Commons (CC BY 3.0) license; note in particular that it is freely available for use in their course Real Analysis II.

Site Licenses: No institution has yet purchased a site licenses for this text. Check our list of site licenses to see whether you are a member of a group that has purchased a site license for one of our other texts.

• Larger print, one page per sheet of paper, bind at left.
  • US letter size: I have read and I accept the Terms and Conditions.
  • A4 size: I have read and I accept the Terms and Conditions.
• Regular print, two pages per sheet of paper, bind at top.
  • US letter size: I have read and I accept the Terms and Conditions.
  • A4 size: I have read and I accept the Terms and Conditions.

Donate: The Trillia Group accepts donations from people who are otherwise licensed to use Mathematical Analysis II free of charge. Donations of as little as US$2 or €2 help us continue to produce and distribute quality online texts.

Errata: There is a list of errata that records changes made to the text since the version of December 18, 2009. The latest version of the book is always available at the links above.