This book is based on the lecture notes of the genius Chinese mathematician and Fields Medalist Tao Zhexuan who teaches a course on practical analysis at the University of California, Los Angeles, and has been loved by readers since the first edition was published. The original book was divided into two volumes, which were combined in Chinese translation.
The book begins with the source of analysis, the structure of number systems and set theory, then leads to the basics of analysis, then moves on to power series, multivariate differential calculus, and Fourier analysis, and finally introduces the Lebess integral, which is almost entirely set in concrete real lines and Euclidean spaces, and perfectly combines rigor and intuitiveness.
This book will introduce advanced real analysis, which is the analysis of real numbers, real sequences, real series, and real value functions. Although real analysis is related to complex analysis, harmonic analysis, and functional analysis, it is different from them. Complex analysis is the analysis of complex numbers and complex functions, harmonic analysis is the analysis of harmonic functions (vibrations), such as sinusoidal vibrations, and how these functions construct other functions through Fourier transforms, and functional analysis focuses on functions (and how these functions construct things like vector spaces).
Analytics is a discipline that rigorously studies these objects and focuses on making accurate qualitative and quantitative analyses of these objects. Real analysis is the theoretical basis of calculus, and calculus is a collection of computational rules that we use when dealing with functions.
The following is the foreword to Tao Zhexuan's Analysis (3rd Edition).
来源 | 《陶哲轩实分析（第3版）》
作者 | [澳]陶哲轩(Terence Tao)
Translator | Li Xin
The content of this book is based on the lecture notes I used in 2003 when I taught a series of courses on advanced real analysis to undergraduates at the University of California, Los Angeles. Undergraduate students generally consider real analysis to be one of the most difficult courses to take, not only because students are exposed to many abstract concepts (e.g., topology, limits, measurability, etc.) for the first time, but also because of the high rigor and proof requirements of this course. Aware of the difficulties of learning this course, teachers are often faced with two difficult choices: either to reduce the rigor of the course to make learning easier, or to adhere to the rigorous standards of the course, but this will make it difficult for most undergraduates, including those who are both intelligent and enthusiastic to learn, to read the material.
Faced with this dilemma, I tried to take a slightly unusual approach to teaching this course. As is usually taught, the introductory part of real analysis assumes that students already have a good understanding of real numbers, mathematical induction, elementary calculus, and the fundamentals of set theory, and that they will quickly move on to the core parts of the course, such as the concept of limits. Normally, when students learn the core content, the textbook will introduce the necessary prerequisites, but most textbooks will not discuss these prerequisites in detail. For example, while students can visually imagine and algebraically perform algebraic operations on real numbers and integers, few students are able to actually define real numbers or integers. In my opinion, this is really a really missed opportunity. Real analysis, linear algebra, and abstract algebra are the first three courses that students take. Through the study of actual analysis, students can truly appreciate the subtleties of rigorous mathematical proofs. As such, this course provides us with an excellent opportunity to review the fundamentals of mathematics and, in particular, to properly and comprehensively grasp the essence of real numbers.
Therefore, this course will be carried out in the following manner. In the first week, I will give some of the more famous "paradoxes" in analytical theory. In these paradoxes, the standard rules of analytic theory (e.g., the commutative law of limit operations and sum operations, or the commutative laws of sum operations and integral operations) are applied in a less rigorous way, leading to some absurd conclusions, such as 0 = 1. This inspires us to go back to the beginning of the course, even to the definition of natural numbers, and requires us to verify all the basic theories from scratch. For example, the first assignment given to students is to prove (using only Piano's axioms) that for all natural numbers, the associative law of addition holds (i.e., for any natural numbers a, b, c, (a + b) + c = a + (b + c), see Exercise 2.2.1). So, even in the first week of the course, students must use mathematical induction to write a rigorous proof process. After deriving all the basic properties of natural numbers, we will start learning about integers (integers were originally defined as the formal differences of natural numbers). Once students can demonstrate all the basic properties of integers, we will begin to study rational numbers (rational numbers were originally defined as formal quotients of integers). Then, we learn about real numbers (through the formal limits of the Cauchy sequence). Along with the above, we will also learn some of the basics of set theory, such as the elaboration of the uncountable properties of real numbers. It is only after learning all of the above (about ten lectures) that we begin to learn what is commonly thought of as the core parts of real analysis: limits, continuity, differentiability, etc.
Learning in this way, the feedback from students throughout the learning process is very interesting. In the first few weeks, students find the material to be very simple on a conceptual level, as only a few basic properties of the standard number system need to be mastered. But on an intellectual level, the teaching material is very challenging. This is due to the fact that in order to rigorously derive higher-order properties from the more primitive properties of the number system, we analyze the number system from the most basic point of view. One student once told me that it was difficult for him to explain two questions to his friends who had not taken advanced real analysis courses: (a) why students in non-advanced real analysis courses were already learning how to distinguish between absolute and conditional convergence of graded numbers while they were still learning how to prove that rational numbers can only be positive, negative, or zero (exercise 4.2.4), and (b) why they felt that their homework was more difficult than their classmates'. Another student told me with great distress that although she was very clear about why a natural number n divided by a positive integer q yielded a quotient a and a remainder r less than q (Exercise 2.3.5), it was very difficult for her to prove this fact, which frustrated her. (I told her that in the follow-up lessons, some propositions were not obvious to be true, and that she would definitely learn to prove them.) However, she didn't seem relieved by what I said. However, these students still enjoy doing homework very much because they are very satisfied by their tireless efforts to give a rigorous proof of an intuitive fact, which reinforces the connection between the abstract processing of normative mathematics and the non-normative intuition of mathematics (and the real world). When asked to give the disgusting "ε − δ" proof of real analysis, they have developed intuitive concepts through extensive practice and have recognized the subtleties of mathematical logic (e.g., the difference between "arbitrary" and "existent" formulations), so that they can easily transition to the "ε − δ" proof, and we are able to go through the course quickly and deeply. By the tenth week, we had caught up with the non-advanced real analysis course, and the students had begun to verify the variable substitution formula for the Riemann-Stillgers integral, and to prove that piecewise continuous functions were Riemannian integrable. By the end of the twentieth week, we had learned (through lectures and homework) the convergence theories of Taylor and Fourier series, the inverse and implicit theorems of multivariate continuous differentiable functions, and established the controlled convergence theorem for Lebess integrals.
In order to get the most out of this material, many of the key foundational conclusions are left as homework assignments for students to prove on their own. In fact, this is a very important point of the course, as it guarantees that students really grasp these important concepts. This pattern will be carried over throughout the study of the book. Most of the exercises are to prove lemmas, propositions, and theorems from textbooks. If you wish to use this book to learn about actual analysis, then I strongly recommend that you do as many of these exercises as possible, including those whose conclusions seem to hold "obviously". The subtleties of this course are not something that can be grasped by simply reading. At the end of most of the chapters in this book, there are a large number of exercises for everyone to learn.
For professional mathematicians, the pace of the book may be a little slow, especially in the first few chapters, with an emphasis on rigor (except for those that are explicitly marked as "informal") and an argument for steps that are generally considered to be obvious and can be passed by. The previous chapters (through tedious proofs) give many of the properties that "obviously" hold in the standard number system, for example, that the sum of two positive real numbers is still positive (Exercise 5.4.1), or that there must be a rational number between two unequal real numbers given at any given time (Exercise 5.4.5). These foundational chapters also emphasize the non-circular argument. The so-called non-circular argument means that the more advanced knowledge that follows cannot be used to prove the earlier elementary theories. In particular, ordinary algebraic arithmetic algorithms cannot be used until they are deduced (in addition, it is necessary to prove that algebraic arithmetic arithmetic laws are true in natural, integer, rational, and real numbers). This is done so that students learn to make abstract inferences using the given finite conditions and derive the right conclusions. Continuing to practice this will help students to use the same reasoning skills to grasp more advanced concepts (e.g., Lebegus integrals) later in their studies.
Because this book is based on the lecture notes that I used when teaching the practical analysis course, it is mainly developed from a pedagogical point of view, and many of the key materials are included in the exercises. In many cases, I have resorted to a lengthy and tedious but educational proof process in place of plain abstract proofs. In deeper textbooks, students will find that the material is shorter, the concepts more concise, and the book is more intuitive than rigorous. However, I think it's important to first understand how to rigorously "hands-on" analysis, as this will help students better grasp modern, intuitive, and abstract analytical methods in graduate school and beyond.
This book places a strong emphasis on rigor and formality. However, this does not mean that all courses that use this book as a textbook should be carried out in this way. In fact, in the course of teaching, I show students more visual concepts (drawing some non-normative graphics and giving some concrete examples) to complement the formal content of the book. Exercises that are set as homework are an important bridge between intuitive images and concepts, and they require students to combine intuitive images with formal understanding to help students argue the topic correctly. I find this to be the most difficult task for students because it requires them to really understand what they are learning, not just memorize what they are learning or just swallow it. However, the feedback I have received from my students is that while homework is a bit of a struggle for them for these reasons, it is also beneficial for them because it allows them to relate the abstraction of normative mathematics to an intuitive sense of basic concepts (e.g., numbers, sets, functions). Of course, the help of a good teaching assistant is also very important in this process.
For the exam section, I would like to suggest one of two approaches: an open-book exam, which can be similar to the exercises in this book (the questions can be shorter and the solution ideas are more conventional), and the other is a homework-style quiz that includes more complex questions. Because the content of the actual analysis is very broad, students should not be forced to memorize definitions and theorems. Therefore, I do not recommend a closed-book exam, nor do I recommend an exam that is done by regurgitating the content of the book. (In fact, during the exam, I provide students with a supplementary sheet that lists the key definitions and theorems that are relevant to the content of the exam.) Setting the test in a homework-like format can help students review and understand the problems in the assignment carefully and comprehensively (as opposed to using flashcards or similar teaching tools to get students to memorize the content of the textbook), which not only helps students prepare for the exam, but also prepares them for general math research.
Some of the material in this book is secondary to the subject matter and can be ignored if time is limited. For example, set theory is not the foundation of analytic theory like number systems, so the chapters on set theory (Chapters 3 and 8) can be briefly and less rigorously skimmed or used as reading material. The content of the appendix on logic and decimal can be used as an elective or supplementary reading and does not need to be taught in the classroom, and the logic section of the appendix is particularly suitable for use as reading material when teaching the previous chapters. In addition, Chapter 16 (on Fourier series) is not used elsewhere in the book and can be omitted.
For reasons of space, the book is divided into two volumes, the first of which is slightly longer, but if minor material is omitted or omitted, the volume can be divided into about 30 lectures. Volume II will cover the contents of Volume I from time to time, but for students who have already taken the Introductory Course in Analysis through other sources, they can be taught directly to them. The second volume is also divided into about 30 lectures.
01
《Analysis of the Pottery (3rd Edition)》
作者:[澳]陶哲轩(Terence Ta)
Translator: Li Xin
This book is based on the lecture notes of Tao Zhexuan, a genius Chinese mathematician and Fields Medal winner, who teaches a course on practical analysis at the University of California, Los Angeles.
The book begins with the source of analysis, the structure of number systems and set theory, then leads to the basics of analysis, then moves on to power series, multivariate differential calculus, and Fourier analysis, and finally introduces the Lebess integral, which is almost entirely set in concrete real lines and Euclidean spaces, and perfectly combines rigor and intuitiveness.
02
Tao Zhexuan Teaches You to Learn Mathematics
Author: Tao Zhexuan
Translator: Li Xin
Fields Medal winner Tao Zhexuan's mathematical thinking analysis, through the Olympiad exercises, take you to understand the beauty of mathematics.
This book is the work of the internationally renowned mathematician Tao Zhexuan when he was 15 years old, which analyzes mathematical problems from the perspective of teenagers, mainly intellectual puzzles such as mathematics competitions, and explains the thinking process in the language of students, which completely shows the problem solving ideas of the teenager Tao Zhexuan.
03
Complex Analysis: A Visual Approach
Author: Tristan Needham
Translator: Qi Minyou
This book openly challenges the current dominance of purely symbolic logical reasoning by explaining the theory of primary complex analysis with a truly unusual, creative perspective and visible way of argumentation.
This book is a book that has had a wide impact in the field of complex analysis. The author has created a unique way to show various concepts, theorems and proof ideas with rich illustrations, which is very easy for readers to understand and fully reveals the mathematical beauty of complex analysis.
04
Introduction to Mathematical Analysis (Iwanami Definitive Version)
Author: [Japanese] Sadaharu Takagi
Translator: Feng Su, Gao Ying
An immortal masterpiece of Japanese mathematics, the "mathematical bible" that nurtured a generation of mathematicians such as Kunihiko Kodaira and Kiyoshi Ito
A masterpiece of analysis written by Japanese mathematician and "father of modern Japanese mathematics" Sadaharu Takagi.
It is known as the "immovable foundation" of the development of modern mathematics in Japan, and has also become the reference point for all calculus textbooks and monographs in Japan.
05
Introduction and Application of Functional Analysis
Author: [plus] Erwin Kresziger
Translators: Jiang Zhengxin, Lu Shanwei, Zhang Shiqi
It is an excellent primer for functional analysis and learning, and is widely used as a textbook for undergraduate and graduate students in the Department of Mathematics and Physics in many universities in Europe and the United States.
Concise, low threshold, with answers, self-study, recommended for the majority of engineering students.
This article is reprinted with permission from the WeChat public account "Turing New Knowledge".
Special Reminder
1. Enter the "Boutique Column" at the bottom menu of the "Huipu" WeChat official account to view a series of popular science articles on different themes.
2. "Back to Park" provides the function of searching for articles by month. Follow the official account and reply to the four-digit year + month, such as "1903", to get the article index in March 2019, and so on.