PDF Ebook Fundamentals of Real Analysis (Universitext), by Sterling K. Berberian
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Fundamentals of Real Analysis (Universitext), by Sterling K. Berberian
PDF Ebook Fundamentals of Real Analysis (Universitext), by Sterling K. Berberian
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"This book is very well organized and clearly written and contains an adequate supply of exercises. If one is comfortable with the choice of topics in the book, it would be a good candidate for a text in a graduate real analysis course." -- MATHEMATICAL REVIEWS
- Sales Rank: #2569345 in Books
- Published on: 2008-06-13
- Original language: English
- Number of items: 1
- Dimensions: 9.25" h x 1.12" w x 6.10" l, 1.55 pounds
- Binding: Paperback
- 479 pages
Review
"This book is very well organized and clearly written and contains an adequate supply of exercises. If one is comfortable with the choice of topics in the book, it would be a good candidate for a text in a graduate real analysis course." -- MATHEMATICAL REVIEWS
Most helpful customer reviews
18 of 19 people found the following review helpful.
at last a clear explanation of integration
By mathwonk
As a student in the 1960's I recall thinking Berberian's book on Hilbert space made it almost too clear, like cheating to read it. I could not understand why no one else wrote like that, not leaving big hurdles for me to jump over. I was taught measure and integration in the most abstract possible way, even Lebesgue measure was hardly mentioned, except as a special case of the largely vacuous looking theoretical machinery we were memorizing without any motivation. All my career I felt left out in the cold from real analysis which I regarded as an impenetrable subject, dry, boring, and useless. Finally as a calculus teacher I began to ask myself why the fundamental theorem of calculus for Riemann integration had a continuity hypothesis instead of just integrability. I gradually learned about Lipschitz continuity (the bounded version of absolute continuity), and began to appreciate the meaning of the Radon Nikodym theorem.
I think now that I would have always understood these matters from the beginning if i had been taught from this book instead of a purely abstract (Halmos type?) approach. Berberian presents Lebesgue measure first, and immediately generalizes it to abstract spaces with a sigma algebra of subsets, instead of the other way around. With this lucid motivation, you say, oh that's why those definitions are made! In regard to Riemann integration, the full story is clearly told in section 5.13. In fact there is only one direction of a satisfactory fundamental theorem for Riemann integration, it is still unknown what intrinsic property of a function, stronger than Lipschitz continuity, characterizes indefinite Riemann integrals, i.e. antiderivatives. I do not know of another book that makes the elementary calculus so clear, or even states this result at all. I love this book. Recommended to any student who wants to understand the subject. These topics may occur in other books, but not so that I grasped them. This author is not trying to prove how clever he is, just how clear the subject can be.
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