NB. I was sent this book as a review copy.
While this book is called An Introduction to Analysis, it contains far more than one might expect from a book with such a title. Not only does it include extremely clear introductions to algebra, linear algebra, intregro-differential calculus of many variables, as well as the foundations of real analysis and beyond, building from their topological foundations, the explanations are wonderfully clear, and the way formal mathematical writing is shown will give the reader a perfect guide to the clear thinking and exposition needed to go on to further areas of mathematical study and research. I think that for an undergraduate student, taking a year to really get to grips with the content of this book would be absolutely doable and an extremely valuable investment of their time. While a very keen student would, I think, be able to go through this book by themselves, as it truly is wonderfully self-contained, if it were used as part of a one year course introducing mathematics in a formal way, I think that this really would be the ideal textbook to cover the foundations of mathematics.
The book covers:
- Algebraic fundamentals
- Topological fundamentals
- Linear Mappings
- Geometry of Mappings
- Differential forms
Certainly at UCT, this would cover a good 50% of the topics which are covered in an undergraduate course, and while some of the topics are covered in more depth than in this book, as a very clear overview, I’ve not seen a better textbook. This would, I suggest be an excellent book for second year students, who have already had a foundational course in integro-differential calculus and basic linear algebra. The abstract level of the book would probably be too much for most students coming straight from high school maths, but I think that the explanations are self-contained enough, that a student who was well motivated, would be able to go through it even in first year and get a huge amount out of it.
The questions in the book are well-chosen, and are usually structured in a sensible way, which takes the reader from a relatively easy set, to a somewhat more challenging set, and generally build nicely from the examples in each section.
Overall, this is one of the best books at the undergraduate level that I’ve ever read, and I have hugely enjoyed many of the explanations of topics which I am familiar with, but may not have studied in this way formally before.