It can be used as a first proof-based course coming after a linear algebra or possibly concurrently. Different people will find it fitting their curriculum differently. But this book provides a very nice two semester course that starts by introducing set theory and induction for the first time and ends with students ready for topology, measure theory, or more advanced calculus.
The clarity of the writing is appreciated. As a proof-based math text of course students will need to be guided through reading it. But the pointers and language are arranged to maximize usability by a faculty person. Because the book has some volume, an instructor can challenge their students to read some sections independently with confidence that the material is arranged clearly enough, obscured only by the usual challenges related to understanding analysis.
This book sticks to its organizing principles. The definition-example-proof-theorem-exercise setup is tried and true. Sections are broken into subsections in appropriate places. It is appropriately self-referential in a way that is necessary for a book building up a mathematical theory. The way in which some sections are optional is explained on the first page of the introduction.
The text builds up the material in a sensible fashion. Some of the ordering choices are the subject of lively pre-existing lively debates. But all of the organizational choices here are logical and lead to a workable year if taken on order. I recommend this book very highly. You should give it a try.
Spaces: An Introduction to Real Analysis
This text covers all the standard material for a senior level undergraduate or master level real analysis class. It start with basics set theory and the real numbers. Then it develops supremum and infimum of bounded sets; limit, limsup, liminf Then it develops supremum and infimum of bounded sets; limit, limsup, liminf for sequences; series of numbers and functions; continuity and uniform continuity of functions, derivative of functions, Riemann integral and improper integrals, basics of metric spaces including connectedness, compactness set and continuity of functions between metric spaces.
Finally it ends with a proof of fixed point theorem. It includes a reasonable number of problems and examples. The text provides an effective index at the end. It is as modular as one can expect of a book on this subject. Since everything numbered consistently one can easily find the results needed from the previous chapters. The material is presented in a logical order.
- Basic Analysis I: Introduction to Real Analysis, Volume I;
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Definitions, theorems, examples, exercises, etc are all numbered in a consistent manner. The book is written in a little bit informal language but that is not a shortcoming. In general, the interface of this book is very typical of an advanced math textbook.
All the statements are numbered and hyperlinked making navigation very easy. I have used it twice in my classes and have been very happy about it.
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The author maintains an errata on his website and has been updating the text regularly. I suggest using the latest edition that can be obtained from the authors website. This is a course in real analysis for those who have already met the basic concepts of sequences and continuity on the real line.
Introduction to Real Analysis - Wikiversity
Here we generalize these concepts to Euclidean spaces and to more general metric and normed spaces. These more general spaces are introduced at the start and are emphasized throughout the course. A comprehensive pack of lecture notes will be provided. The following may prove useful:.
MAT 266 - Introduction to Real Analysis
Teacher responsible Prof Martin Anthony. Topics covered are: Sequences and series on the real line.
Metric and normed spaces; open and closed sets, topological properties of sets and equivalent metrics, sequences in metric spaces, compactness, completeness.