Real Mathematical Analysis (Undergraduate Texts in Mathematics) 2nd Edition by Charles Chapman Pugh (PDF)

Description

Based on an honors course taught by the author at UC Berkeley, this introduction to undergraduate real analysis gives a different emphasis by stressing the importance of pictures and hard problems. Topics include: a natural construction of the real numbers, four-dimensional visualization, basic point-set topology, function spaces, multivariable calculus via differential forms (leading to a simple proof of the Brouwer Fixed Point Theorem), and a pictorial treatment of Lebesgue theory. Over 150 detailed illustrations elucidate abstract concepts and salient points in proofs. The exposition is informal and relaxed, with many helpful asides, examples, some jokes, and occasional comments from mathematicians, such as Littlewood, Dieudonné, and Osserman. This book thus succeeds in being more comprehensive, more comprehensible, and more enjoyable, than standard introductions to analysis.New to the second edition of Real Mathematical Analysis is a presentation of Lebesgue integration done almost entirely using the undergraph approach of Burkill. Payoffs include: concise picture proofs of the Monotone and Dominated Convergence Theorems, a one-line/one-picture proof of Fubini’s theorem from Cavalieri’s Principle, and, in many cases, the ability to see an integral result from measure theory. The presentation includes Vitali’s Covering Lemma, density points — which are rarely treated in books at this level — and the almost everywhere differentiability of monotone functions. Several new exercises now join a collection of over 500 exercises that pose interesting challenges and introduce special topics to the student keen on mastering this beautiful subject.

User’s Reviews

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

⭐I had attempted to study real analysis through baby Rudin, but found it too terse. In contrast, Pugh’s book is delightfully lucid, neither too concise nor overly wordy. He uses pictures to help readers develop a more intuitive understanding of analysis, which is incredibly helpful for beginners. There are plenty of examples and interesting exercises, making this book even more suitable for someone aiming to self-study real analysis. The chapter on topology, as other reviewers have noted, is especially illuminating. I also particularly enjoyed the chapter on function spaces.Some of the arguments felt slightly handwavy at certain points (most notably, the proof on the Implicit Function Theorem), but I found that the gaps were easily filled in by the reader. Overall, this was an excellent (and surprisingly affordable) book that made me fall in love with analysis.

⭐Appears to be a defective “Print on Demand” hardcover edition: the ink is too light to be able to read many of the pages. This text is supposed to be great, and I wish I could have been able to find out for myself.

⭐This textbook serves as an excellent introduction to real analysis. There are helpful examples and illustrations throughout, and the author has a good sense of humor. Some of the concepts can be a little tricky to make sense of on the first read through.

⭐Great text book in basic analysis.

⭐Surprisingly clear instruction and elaboration of some concepts I found very difficult.

⭐(There are already several reviews about the content of the book under the first edition.)During the last year, Springer has published new editions of several books in the UTM series (Axler: Linear Algebra Done Right, Ross: Elementary Analysis, Abbott: Understanding Analysis). I have bought all of these from Amazon and received nicely produced books, on good paper quality and all in exactly the same format.When I received the second edition of Pugh: Real Mathematical Analysis, however, it was immediately clear that this is a print-on-demand book, even though it was first published on July 30, 2015. The format is weird, much bigger than the other books in the series. The paper quality is cheap, like what you use for your printer, not what you expect from a book.Presumably Springer has not even bothered to do a first run of proper copies, but have gone straight to the POD quality that they use for most books. I thought it was safe because the book was just published, but I was wrong. Maybe I was just unlucky, but I just wanted to warn other buyers who might care about this issue.

⭐Incredible book. This is how a top-notch mathematicians writes when he actually cares about educating the reader and aren’t lazy. When they decide, “Maybe I can be a good teacher and not pretend like all of this was a cakewalk.” It’s also proof that a rigorous math text doesn’t have to be dry and diagramless. I own many calculus and real analysis books. From Spivak, Apostol 1 & 2, Morrey, Hille, Rudin, Gleason, Loomis and Sterberg, Stromberg, Bishop, Thomas, etc. This book holds its own, and for around $25.00, it’s a steal. I’m used to authors that basically throw you in the deep-end; if you drowned, so be it. You weren’t meant to be a mathematician. Professor Pugh gets an “A+” for effort and content with this book. It’s beautiful and should be required reading in all college calculus courses. For any students that pick this book up and complain, you shouldn’t be a mathematician or in a deductive discipline. Take up acrostics or beekeeping, as Kleene’d likely put it.

⭐I don’t mean to be mean, but I think most people write books to share knowledge, but a principle of good teaching is first you gie a concrete example and then work your up to abstractions. I don’t see that here. There are blanket statments made, and perhaps they are obvious to the author, but not to the student. Using words like “Totally natural” is not helpful. What’s natural or not depends on how long one has been studying this material.

⭐This book is a pleasure to read, and contains some of the best sections of topology related materials. While there are some excellent and standard exercises spanning over several pages… some of them appear out of the blue in difficulty that would often be solved using methods of functional analysis. If you have a professor that can guide you through exercises and pick out those appropriate then the book is excellent – but if not you could be stuck down a rabbit hole on some of them never to return.

⭐J’aurais plutôt donné une note de 4,8!.J’ai reçu ma commande très bien emballée et en condition (on ne peut plus) parfaite , cependant, les délais de livraison ont dépassé la date estimée de réception la plus tardive de quelques jours et on ne pouvait suivre le colis à la trace (code non-transmis à amazon d’après les informations disponibles sur ce site). À part ces deux petits anicroches, tout est bien qui finit bien!L.Chénard

⭐

⭐The book seems like a good introduction to real analysisThe book doesn’t like to be opened. Even now the cover won’t go past a point about halfway down. The back is broken and several pages on the front have fallen out.

⭐O iivro e’ o’timo!!

⭐

⭐Hard, rigorous, but great book

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