Mathematical Analysis II
by Elias Zakon
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.
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- Larger print, one page per sheet of paper, bind at left.
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This text is appropriate for any second course in real analysis or
mathematical analysis, whether at the
undergraduate or graduate level.
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 December 18, 2009.
Terms and Conditions: Thanks to funding from the Saylor Foundation, Chapter 6 is licenced under a Creative Commons Attribution 3.0 Unported (CC BY 3.0) license. Chapters 7, 8, and 9 are licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0) Public License. If you would like more rights to this work (e.g., commercial publication rights or including parts of this work in another work, etc.), please contact the publisher.