Analysis I: Third Edition (Texts and Readings in Mathematics)

This book is the best introduction to Real Analysis/Advanced Calculus that I have seen so far. Unlike the terse Baby Rudin (i.e. Principles of Mathematical Analysis by Rudin), this book has paragraphs upon paragraphs on the motivations behind definitions and concepts. And each next step is laid out naturally following the previous step; plus the steps are small enough for a beginner Mathematician to follow. Like Baby Rudin, this book is one of very few that I have seen which rigorously constructs the Number Systems – i.e. the natural numbers, the integers, the rationals and the reals – in that order. In that regard it surpasses Baby Rudin as the latter only contains the construction of the reals. Also while there are many exercises, the author makes sure to give copious hints on the harder ones. Again, this will be in great service to any beginner.

I like the whole Tao’s idea for being sometimes informal in Analysis themes. This will help undergraduate students to go far away! Certainly they have to work hard with the proofs even though many hints are given.

College book

As expected, this is a well written book on not an easy subject. As a non-professional mathematician, the book offers a very good introduction to the subject, with proper explanations and examples. For the dedicated student, it is an excellent textbook. For the amateur, it is better to read it one time "on the fly", to get a feel for the subject, and then come back to deal with examples and exercises. Overall, a great text on a complex subject.

More comprehensible than the legendary baby rudin book.

wunderbar

This is the best Mathematical Analysis book I know.Wonderful!!!

As described, thanks

One page was stuck together, but overall quality is acceptable for the price. Tao's writing is very clear and offers sufficient detail to self-teach analysis.

It's hard to overestimate how good this book is. Don't be fooled to think starting from the basics takes out a lot of the flavor of analysis. Indeed, I personally realized how lacking my foundations were as I was taken thorugh a tour disecting every number system. After your perception about these number systems has dramatically shifted, the transition to the core of the subject flows smoothly. More importantly, you always get the feeling of being in good hands: there's not a single argument seem to be taken out of thin air because Tao makes the effort of going the extra mile in terms of rigour. For example, he first defines what it means for a sequence to be epsilon-close to a real number, what it means to be eventually epsilon-close, and then what it means to converge. That kind of scrutinizing is present across the entire book (see how he defines the Riemann integral by first defining the piece-wise constant Riemann integral instead of dropping the traditional upper and lower Riemann sums). It's because of these extra layers that every concept flows smoothly.When it comes to the exercises, most consist in proving main results from the text. This makes you engage with the subject at a deeper level. It may sound overkill, but it is really rewarding. There are also those that expand beyond the body of the text, those that ask you to come with examples, and counterexamples, etc. There are no solutions but in my opinion, there are enough hints to not get stuck. I believe that as one progresses in mathematics hints are more valuable than complete solutions. More importantly, every exercise serves a purpose and educates the reader in how one should read a math textbook: fighting it! In the same way he often inserts a "why?" inside the main text, Tao wants you to learn how to read mathematics and think about mathematics, questioning every step in the way.Why isn't this textbook more widespread? Why math departments keep throwing beginners a slog like Baby Rudin? I wish I could tell, because given the modern alternatives like Tao's, I wouldn't look any further.

Wonderful text, just like the sequel. Starts with an axiomatic approach to define the number systems. Introduces cauchy sequences to define the reals, then takes a closer look at limits, contunuity, differentiation and integration. So it's absolutely ideal as a self contained volume for first year analysis.The sequel resets things in a metric space and looks at uniform convergence and power series, then the usual primers for multivariate calculus and lebesgue integration. However you will need Rudin to set up differential forms, but the book does include some other niceties, like a warm little chapter on fourier transforms and a few other nice little nugets interspersed throughout.The publishers etc. are another story altogether, though. They overshot the shipping deadline of the first volume, and then tried to stall by gaslighting me about not having provided a proper address. Several books in the same order, with the same address, were already on the desk in front of me. When I switched it back on them by pointing this out and threatening to pull the money, they went quiet and sent the book. However, the book that arrived manages to go from p158 to p335 in the appendix, then full back matter, before returning to p159 and continuing on. In short, they are flaky AF and it's a real shame such an essential mathematics series is only available through them.

Ottimo testo di Analisi reale che spiega i concetti e le loro connessioni e posto in una impalcatura oltre a presentare il debito sfilare di definizione- proposizione-corollario-lemma. Snello, con esempi e controesempi perfettamente mirati allo scopo didattico, pochi esercizi, nessun fronzolo o quasi. Didattico e accessibile nel senso che non moltiplica la difficoltà della materia con la cripticità ingiustificata o con la prolissità in buona fede che si trova in tanti testi didattici

A really nice book on analysis. The book reflects the author's passion for mathematics. Easily it is one of the best introduction to real analysis (most probably the best). The book requires no prerequisite knowledge of the topic and develops the topic from scratch. A must have for any maths enthusiast/student. Rest you can judge after buying and reading the book itself, you will easily fall in love with the writting style of the book. I have attended the photo of index so that you can check it's contents.