Is "Foundations of Mathematical Analysis" by J.K. Truss a good introduction to analysis?

Question

I bought the book at some old bookstore, and I read the first few pages and it seems quite interesting. I want to learn analysis, but I am not sure if this book is the most appropriate introduction. I was thinking of doing Baby Rudin instead. What do you recommend?

Answer 1

From the publisher:

Foundations of Mathematical Analysis covers a wide variety of topics that will be of great interest to students of pure mathematics or mathematics and philosophy. Aimed principally at graduate and advanced undergraduate students, its primary goal is to discuss the fundamental number systems, N, Z, Q, R, and C, in the context of the branches of mathematics for which they form a starting point; for example, a study of the natural numbers leads on to logic (via Gödel's theorems), and of the real numbers (as constructed by Cauchy) to metric spaces and topology. The author offers a refreshingly original and accessible approach, presenting standard material in new ways and incorporating less mainstream topics such as long real and rational lines and the p-adic numbers. With a discussion of constructivism and independence questions, including Suslin's problem and the continuum hypothesis, the author completes a wide-ranging consideration of the development of mathematics from the very beginning, concentrating on the foundational issues particularly related to analysis.

This book is out of print and I haven't read it. Judging from the publisher's introduction, it seems that the word “foundations” in the title refers not to the basics or fundamentals of mathematical analysis, but to the logical foundations upon which mathematical analysis can be rigorously built. (If you have heard about the “three crises in mathematical history” in the past, you know what I mean.) This is probably why topics on mathematical logic or axiomatic set theory, such as Gödel's theorems, Suslin's problem or continuum hypothesis, are discussed.

While the book does touch upon some topics in undergraduate-level mathematical analysis (such as metric spaces), I guess that most topics, notions or theorems that one encounters in a typical analysis text (e.g. the ratio test, mean value theorem, L’hospital’s rule, Lebesgue’s criterion of Riemann integrability, uniform continuity, or uniform convergence) are not covered here. If you want a textbook that covers those conventional topics, you should probably look elsewhere.

I don't know what books suit you the best. Baby Rudin is a well-known hit-or-miss. Many people who like the book say that the first seven or eight chapters are masterfully written. Those who don’t like it criticise that the text is unmotivated and explanations are terse. But most people seem to agree that the last few chapters (especially the one on Lebesgue integral) are not as good as the earlier ones. Personally I love its chapter 2 (basic topology) and I like parts of chapters 7 and 8, but by and large I think this book is grossly overrated.

At any rate, to pick a text that suits you, I think the rules of thumb are:

1. Don’t rush to buy it. If possible, borrow it from the library (or download an electronic copy from the internet if there is one) and read it. If you cannot borrow one, try to find its table of content online and see if its scope is right. Read also its preface. Books that have the authors’ pedagogical approaches or mathematical insights explained in prefaces are often better than the ones that show only flowcharts and acknowledgments. And of course, read also book reviews if there are any.
2. Try to borrow a number of different texts and pick the one(s) you like.
3. If time and/or budget allow, try an easier book first. A highly acclaimed book written by a world-famous author is useless if you don‘t understand it. (In this regard, I think the book Real Variables with Basic Metric Space Topology, written by the late professor Robert Ash and downloadable from his website, is a good place to start.)

Answer 2

I recommend 'Mathematical Analysis Linear And Metric Structures And Continuity'. This is a game changer.