Real and Abstract Analysis: A modern treatment of the theory of functions of a real variable by E. Hewitt (PDF)

Description

This book is first of all designed as a text for the course usually called “theory of functions of a real variable”. This course is at present cus­ tomarily offered as a first or second year graduate course in United States universities, although there are signs that this sort of analysis will soon penetrate upper division undergraduate curricula. We have included every topic that we think essential for the training of analysts, and we have also gone down a number of interesting bypaths. We hope too that the book will be useful as a reference for mature mathematicians and other scientific workers. Hence we have presented very general and complete versions of a number of important theorems and constructions. Since these sophisticated versions may be difficult for the beginner, we have given elementary avatars of all important theorems, with appro­ priate suggestions for skipping. We have given complete definitions, ex­ planations, and proofs throughout, so that the book should be usable for individual study as well as for a course text. Prerequisites for reading the book are the following. The reader is assumed to know elementary analysis as the subject is set forth, for example, in TOM M. ApOSTOL’S Mathematical Analysis [Addison-Wesley Publ. Co., Reading, Mass., 1957], or WALTER RUDIN’S Principles of M athe­ nd matical Analysis [2 Ed., McGraw-Hill Book Co., New York, 1964].

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐Edwin Hewitt was my undergratuate mentor at the University of Washington. He used this then new text for my sophomore real analysis course in the honors curriculum. It was all very exciting and heady! He was a masterful teacher and researcher, always commanding class attention. A faculty Nobel Laureate in physical chemistry even attended. Over the years my original copy disintegrated fromuse, so I was very pleased to obtain a new one from Amazon. The content extends well beyond traditional real analysis and opens wide the doors of harmonic and functional analysis. It has been nice to have my old friend back.

⭐After 30 years of its first publications, this book remains one of the best. It treats all the elements of a graduate course in analysis thoroughly. One of the few books that actually goes through construction of the real number system (in Chapter 1) and products of infinitely many measure spaces (in Chapter 6). There is plenty material to choose from, and all the standard topics are covered. In a very few cases notation is a bit cumbersome (or obsolete), but this is no big problem. Highly recommended to all graduate students in mathematics.

⭐In many applications today, such as PDE or harmonic analysis, it is crucial to have a really good grasp of the Lebesgue integral and absolutely continuous (rather than continuously differentiable) functions. Many real analysis books shy away from these matters.Hewitt and Stromberg provide *every* detail, starting from scratch with measure theory, including Caratheodory’s construction and fine distinctions such as Lebesgue vs. Lebesgue-Borel measurable sets. They proceed to discuss the Lebesgue integral in detail, then differentiation and absolutely continuous functions, the Lebesgue spaces including Riesz representation and a Banach space primer and finally integration on products (also infinite) of measure spaces. Fourier series and transforms are covered as well as fine details of (naive) set theory.The book would be perfect if some standard analysis (Taylor series, analytic functions, Hospital rule, …) was discussed; for this reason and because of its high level, it cannot be recommended as a book for 1st year calculus.

⭐This is the kind of book I appreciate the most: one that’s always got the information you need. This meticulous text covers thoroughly just every topic from elementary set theory up to product measures. It develops carefully all topics that should be included in standard analysis lectures (set theory, topology, Lebesgue integral, Banach and Hilbert spaces, differentiation, product measures) at such level of abstraction that the book turns out to be suitable for introductory courses, advanced courses, and later reference. The only one shortcoming I see is that the book includes no bibliography.Please check my other reviews at my member page (click on my name above).

⭐The book covers all the essential points and abstract structures to fully understand integration and differentiation. I believe it is hard to find another book with such clarity in exposition. Another I like about this book is the carefully designed exercises. For instance, Legendre polynomials are developed in the exercises. This is something you can find Stromberg’s solo analysis book as well. If you are self-taught then you’d find these types of exercises very useful. In fact, compared to other textbooks Hewitt/ Stromberg shines on this aspect. Overall, it is an excellent book to have!

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