Analysis I

by Terence Tao

This is part one of a two-volume book on real analysis and is intended for senior undergraduate students of mathematics who have already been exposed to calculus. The emphasis is on rigour and foundations of analysis. Beginning with the construction of the number systems and set theory, the book discusses the basics of analysis (limits, series, continuity, differentiation, Riemann integration), through to power series, several variable calculus and Fourier analysis, and then finally the Lebesgue integral. These are almost entirely set in the concrete setting of the real line and Euclidean spaces, although there is some material on abstract metric and topological spaces. The book also has appendices on mathematical logic and the decimal system. The entire text (omitting some less central topics) can be taught in two quarters of 25–30 lectures each. The course material is deeply intertwined with the exercises, as it is intended that the student actively learn the material (and practice thinking and writing rigorously) by proving several of the key results in the theory.

Genres: Mathematics, Science, Textbooks, Nonfiction

347 pages, Hardcover

First published January 1, 2006

About the author

Terence "Terry" Tao FAA FRS (simplified Chinese: 陶哲轩; traditional Chinese: 陶哲軒; pinyin: Táo Zhéxuān) is an Australian-American mathematician who has worked in various areas of mathematics. He currently focuses on harmonic analysis, partial differential equations, algebraic combinatorics, arithmetic combinatorics, geometric combinatorics, compressed sensing and analytic number theory. As of 2015, he holds the James and Carol Collins chair in mathematics at the University of California, Los Angeles. Tao was a co-recipient of the 2006 Fields Medal and the 2014 Breakthrough Prize in Mathematics.

Reviews

[Madhusa K]

Tao has taken a very literal approach to teaching by first principles.

It is typical for Analysis books to begin by constructing the Reals. Not for Tao though, he takes a unique approach. He starts at nothing. This is very much a Definition-Therom-Lemma-Proof book, but a large part of that is because he refuses to use anything that he first hasn't constructed from the ground up! After starting at the Peano Axioms to create the Natural numbers Tao moves on to construct what is ZFC set theory then to the integers and finally rationals. But before he constructs the reals he has to first construct the notion of a sequence and a limit.

This is not a reference book. You begin from the start and work your way through page by page. A phenomenal introduction to the subject.

[Kevin Doran]

Grueling. The amount of work required of the reader per page is daunting. You could easily spend an average of 2 days per page on this book.

Once you get accustomed to the pace, it's great. You can't really get lost as it really does start at the beginning and lays out each step meticulously.

Specific issues
1. Despite this being the 3rd edition, there are still a handful of minor errors.

2. Lack of solutions to questions. Solutions are key for self-study in my opinion. There are many answers scattered around the internet, but it is a pain to find them when I was checking my work or stuck on something.

Maybe I'm wrong to give only 4/5, but when I look over at the book on the shelf, it's not really an affectionate feeling I have. More like how you might look at a spin class instructor of a particularly punishing class. I feel like that's not really the optimal vibe for a book.

[Ida]

I did not read through the whole book, I must confess, since I picked it up for the purpose of finding interesting homework problems for my students. I believes it gives a good general impression of things, but it would not be my main choice for learning real analysis.

[Kent Sibilev]

I have a mixed feeling about this book. This is a very rigorous treatment to the mathematical analysts. Unfortunately, most chapters consist of the definitions and theorems without any proofs. I’m guessing It would be absolutely amazing resource for a lecturer or an instructor of the subject, or as a course material when you have a chance to review your homework in the class. But for the self study, you would need to refer to other books.

So if you are new to the math analysis, there are much better books like Abbotts’ Understanding Analysis or Baby Rudin to name a few.

[Zichang]

In a year to fully read the book. Really clear and detailed content written in the book. Expecting to finish the II book, but maybe need a while to absolutely digest the theorems and text in the I one.

[Fumiteru]

The best book that exists to change the perception of a newbie and introduce pure mathematics through the eyes of one of the best mathematicians of our time, Terence Tao. This book wasn't easy to go through, real anaylsis isn't in the beginning, but it was so much different than other introductory anaylsis books and so much better!

[Nick DeSanctis]

Personal favorite book for analysis. Tao makes arguments and proofs easy to digest without hiding the necessary math. Includes metric spaces, a point over on Abbot’s “Understanding Analysis”, but I do find the exercises to be miles easier than those in Baby Rudin. Still, for learning analysis, this is at the top.

[Mynk Pe]

this book is perfect for you if you want a taste of the meticulous mathematical rigor. the author's remarks are very helpful. many important propositions are left as exercises ( but they are not so hard to be proved as there are also quite a lot of hints).

[Sabrina]

this has to be one of the best books I've ever read for mathematics. not only are the problems absolutely amazing, the writing as a whole is absolutely beautiful. more math books should focus on readability! overall Terence tao's analysis 1 was a very wonderful read.

[Mewus Hylian]

the lack of an answer key and enough example solutions makes it hard to compare my solutions with the correct ones and get feedback. its a difficult book for self study unless your parents happen to be math professors

[Liora Sophie]

Terry is too smart for the rest of us but to be honest I did find this book helpful on multiple occasions during infi 1 and 2.
Not enough detail, please elaborate more.

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[Rafał Grochala]

Eye-opening, but also - not gonna lie - too wild in some places. Last chapter (or two) were too fast. Probably the second volume will clear things up.

[Alex Carlman]

odd to say perhaps, but this is a life changing book

[Sangam Yadav]

One of the best book in analysis

[Joseph Bronski]

kind of cured me of my finitism. convinced me reals are kind of useful and at least logically consistent

[Ayan Basak]

great

[Andre]

Love this book so much. It's deadass just lecture notes with exercises.

[Benjamin Seignour]

Terrence Tao has published an incredibly comprehensive textbook that serves as a wonderful introduction to the subject for interested outsiders and undergrad students in mathematics. The work demonstrates Mr. Taos unique ability to combine his other worldly mathematical skill set with writing that is crisp and engaging. Motivating the study of analysis via the construction of the number systems vis vi peano arithmetic, provides the reader with a road map for properly conducting mathematical reasoning in highly abstract and proof based settings. Thereafter, Mr. Tao provides excellent treatment of sequences, series, and continuity. We are then introduced to the world of derivatives and integrals. Along the way Mr. Tao deals in quotes, anecdotes, and mathematical history with prose that you rarely find in mathematics textbooks of this calibre. This book was my introduction to the world of pure mathematics and it served me wonderfully. Mr. Tao truly is not just a mathematical genius, but a fully rounded savant. It wasn’t mathematics that found Tao, but rather that Tao found mathematics.

[Matt]

Tao's ability to explain concepts is unparalleled in the realm of undergraduate-level analysis texts.