Mathematical Analysis: Volume I

This book is the first volume of the textbooks intended for a one-year course in mathematical analysis. We introduce the fundamental concepts in a pedagogical way. Lots of examples are given to illustrate the theories. We assume that students are familiar with the material of calculus such as those in the book Calculus: Ninth Edition 2020 by James Stewart, Daniel K. Clegg, and Saleem Watson. Thus, we do not emphasize on the computation techniques. Emphasis is put on building up analytical skills through rigorous reasonings.

Besides calculus, it is also assumed that students have taken introductory courses in discrete mathematics and linear algebra, which covers topics such as logic, sets, functions, vector spaces, inner products, and quadratic forms. Whenever needed, these concepts would be briefly revised.

In this book, we have defined all the mathematical terms we use carefully. While most of the terms have standard definitions, some of the terms may have definitions differ from author to author. The readers are advised to check the definitions of the terms used in this book when they encounter them. This can be easily done by using the search function provided by any PDF viewer. The readers are also encouraged to fully utilize the hyper-referencing provided.