Introduction to Analysis by Arthur Mattuck (PDF)

Description

This book is meant for those who have studied one-variable calculus (and maybe higher-level courses as well), generally skipping the proofs in favor of learning the techniques and solving problems. Now they are interested in learning to read proofs, and to find and write up their own: perhaps because they will need this for the next steps in their chosen field, or for intellectual satisfaction, or just out of curiosity. There are two paths to this. Some books start with a great leap forward, giving the definitions in n-space. This requires an excursion into point-set topology, whose proofs are unlike those of the usual calculus courses and are a roadblock to many. The path chosen by this book is to start like calculus does, in 1-space (i.e., on the line) and focus on the basic definitions and ideas of one-variable calculus: limits, continuity, derivatives, Riemann integrals, and a few more advanced topics. It’s done rigorously, but also in as familiar a way as possible. So from the start it will use as a source of examples what you know (with occasional reminders): K-12 mathematics and basic one-variable calculus, including the log, exp, and trig functions. This takes up about two-thirds of the book, and might be as far as you wish to go. It sounds like just repeating calculus, but students say that it feels very different and is not all that easy. The rest of the book gets into ideas from advanced calculus used in lower-level courses without proof: uniform convergence, differentiating infinite series term-by-term and integrals containing a parameter (the Laplace transform, for instance). For the latter, it’s finally time to learn about point-set topology in the plane (2-space, but n-space is no harder). There’s also for the curious or needy an optional chapter with the most important facts about point-sets of measure zero on the line and a more powerful integral, the Lebesgue integral. Two appendices respectively provide needed and optional background in elementary logic, and four more give interesting applications and extensions of the book’s theory. For more details, click on “Look Inside” to see the Table of Contents. Some generally helpful features: –Leisurely exposition, with serious comments about proofs, other possible arguments, writing advice; some semi-serious comments too; –Attention paid to layout and typography, both for greater readability, and to give readers models they can imitate; –Questions after most sections of a chapter to firm up what you just read, with Answers of various sorts at the end of the chapter: single words, hints, complete statements, formal proofs. Mathematically helpful features: –The language of limits is simplified by suppressing the N and the delta when their explicit value is not needed in the argument, replacing them with standard applied math symbols meaning “for n large” and “for x sufficiently close to a”. These are introduced carefully and rigorously; some caution is needed, which is described at the end of the Preface (click “Look Inside”). –The book tries to go back to the roots of real analysis by emphasizing estimation and approximation, which use inequalities rather than the equalities of calculus, but have a similar look, so that many proofs are calculation-like “derivations” that seem familiar. But inequalities are often mishandled and warnings are given. For examples of these features and writing style, go to the author’s home page, link to “book”, then link to “sample pages” from the first three chapters. The book was developed at MIT, mostly for students not in mathematics having trouble with the usual real-analysis course. It has been used at large state universities and at small colleges, as well as for independent study. Students evaluate it as readable and helpful. The new printing, by CreateSpace and at a reduced price, is the eighth, incorporating all known significant corrections and a new Appendix 6.

User’s Reviews

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

⭐I used this book for a Real Analysis class at MIT. This book is amazing. The author is a genius, his methodology is perfect for learning. It reminds me a bit of Gilbert Strang’s way of thinking and exposing concepts – another absolute genius on teaching. It’s a great book to study on your own, and if you prove the theorems yourself and work through the Questions without reading the solutions, you’ll very likely learn all the most relevant concepts and be able to tackle something more complex like Rudin. It’s all about understanding the concepts, which is far more important than banging your head with Rudin and not getting anywhere.This was actually my first contact with writing actual math proofs after my undergraduate in engineering, and I think I couldn’t have had a better introduction to this.Yes, the printing quality is not the greatest, as other people have mentioned. I tend to be bothered by this kind of stuff but with this book it’s really not that bad, I stopped noticing it after the first minutes. Highly recommended!!

⭐Does Amazon screen vendors for copyright violation? This book looks suspiciously like a one of several I’ve seen originating from China or Taiwan, or “elsewhere.” The “photocopy” appearance of the print is the most telling example of this behavior.The content, as many reviewers have mentioned, is fine.

⭐This is an introductory text on real analysis that will prepare the reader well for further reading. The discussion is at a very elementary level, but no less useful for all that. Mattuck’s sense of humor glimmers throughout the text (‘Theorems are there to save work. Adults cite theorems.’); he was my second-term lecturer in calculus at MIT, and the humor that was present in his quizzes is evident here. Highly recommended for someone who wants a gentler introduction to analysis than typically provided by books such as Rudin. I recommend also referring to Course 18.100A, Introduction to Analysis, on the MIT Open Courseware site, which uses this book.Mattuck’s book will not cover some topics that are found in a more traditional analysis book such as the implicit function theorem, continuity defined in terms of open sets, Lebesgue integration (touched on only briefly here), and Stokes’s Theorem. Nonetheless, a very useful text and a bargain at the price.Some reviewers have complained about the print quality, but I don’t find that to be much of a problem.

⭐The print quality is low and hard to read. The actual topics covered are low quality as well. The author uses a lot of non-standard notation and spends many pages to create mountain out of a molehill. Abbott (and several other books) strictly dominate this book. What this means is: there is never a good reason to buy this book.

⭐Really easy to read, and a great intro. I am thoroughly enjoying it and learning so very much from it! It’s a great compromise between learning “higher level math” and still being an introductory book. I wish I had discovered this earlier. Also, it’s amazing that this textbook isavailable for so cheap. I wish all textbooks were so accessible (both financially and read-ably). Maybe the world would be a smarter place.

⭐Print quality is poor. Very low resolution.

⭐An excellent book for explaining concepts and proofs. I find it extremely helpful supplementing another text book I am presently using.

⭐The book is in good quality. No scratches inside. The hard covers are good.

⭐Seriously, it doesnt really get any better than this in a mathematics textbook especially for self study. Very clear and the end of mini chapter questions are cool and refreshing!

⭐The content of the book is good, but the quality is terrible.

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