The Trilogy of Mathematical Analysis: Part One

Principles of Mathematical Analysis, Third Edition, known in Chinese as 《数学分析原理，第3版》.

This book, commonly referred to as PMA, is one of the most classic textbooks in mathematical analysis. It is ideally suited for students who have mastered calculus and linear algebra and wish to advance into pure mathematics. The content is extremely rigorous, abstract, and concise, making it quite challenging.

The author, Walter Rudin, holds a PhD in pure mathematics from MIT, with research focuses on complex analysis, functional analysis, and harmonic analysis. After graduation, he served for many years as a faculty member at the University of Wisconsin–Madison. Renowned for his humility, rigor, and high mathematical expertise, Rudin taught mathematics with the artistry of an art class and was highly popular among students. However, students often complained about the difficulty of his texts. Rudin authored three textbooks that have become timeless classics: in addition to the one introduced above, there are Real and Complex Analysis and Functional Analysis. Due to the wide influence of these three works, many universities worldwide adopt these textbooks for teaching mathematical analysis. Colleagues in the mathematics community have affectionately nicknamed these three books “Baby Rudin,” “Big Rudin,” and “Functional Rudin.” This not only reflects their ascending levels of difficulty but also serves as a convenient shorthand—“Baby” highlights the introductory nature of Principles of Mathematical Analysis, “Big” emphasizes the broader scope of Real and Complex Analysis, and “Functional” underlines the advanced depth of Functional Analysis.

Although Rudin’s textbooks are challenging, mastering them instills deep appreciation for his rigorous approach. For readers, the true benefit comes not merely from reading but from independently working through Rudin’s exercises and proofs. This rigor trains readers to think both intuitively and efficiently.