Elementary Classical Analysis

This was my favorite book as an undergraduate student and I've taught from it as a professor. It is an excellent geometric approach to analysis. It can even help students who have difficulty with epsilon delta proofs understand the geometric intuition behind them. The construction of the real line at the beginning is daunting for students who aren't clear about set theory and sequences already but a few supplementary materials can help the students out there (see my webpage notes on real analysis for example). The proofs are hidden which makes it a challenge for students to try prove everything themsleves before peeking at them, but they are available. Just remember to tell your students where they are! As a student I loved the book because it allowed me to learn everything on the metric space level while allowing students who prefer to stay in Euclidean space to do that. Now I am a metric geometer.

I am using this book to teach myself analysis. Because my mathematical background is limited, I cannot assess what the book is missing, or whether alternative methods of presentation would be more insightful. But in terms of clarity and comprehensibility, the book does very well. The authors write very carefully and are not cryptic; the proofs and examples are well-presented, and I rarely feel lost. The book is rigorous but not, let's say, snobbish. I am learning a lot from it.

Many other reviewers have panned this book. The overall sentiment seems to be that the book is too difficult to follow. Perhaps for them. And, granted, perhaps this is so for many readers. But some students, who are probably majoring in maths or physics and who might be amongst the top in their classes, are likely to appreciate the book. It is a rigorous explanation of classical analysis. Frankly, for someone who will not major in maths, you are unlikely to need this level of rigour in your understanding and usage of the maths. Even theoretical physicists. But you can regard it as a good part of your maths education. If you have learnt introductory calculus at the level of Apostol or Spivak's books, then that level of rigour is continued here. The proofs can be quite difficult to follow. It is for good reason that Marsden segregates these into the ends of the chapters. The fact that these proofs are difficult is perhaps misread by some reviewers as a flaw in Marsden's writing. Wrong. Some proofs are inherently difficult, and need a detailed and careful presentation. The Heine-Borel Theorem, for example. Which is why I find puzzling claims by some reviewers of many errors in the text. Are they referring to simple typos? Or errors in the logic? If the latter, maybe they should cite specific cases. I went through an earlier edition, as a student, and studied carefully most of the proofs. Beyond some typos, I never found any logic errors.

I used this book (earlier Edition) in one of my Analysis in Rn courses at UChicago. Previously I had used 1st Ed. Wade and was very disappointed with its lack of explanatory clarity. M & H's book proved much more clearly written and with better examples. It was indispensable in supplementing the lecture material and facilitating self-study later. Using this book was the first time I ever realized that I could enjoy a Math Textbook, and furthermore feel confident enough to study independently without course lectures or a professor to fill the gaps. I originally borrowed my copy from the Math Department and had to return it later. I think it may be time to pick another copy up.

Marsden and Hoffman have done an admirable job combining clarity and rigor in a book appropriate to the level of an advanced undergrad class at a good university. The organization and tone of the work set it apart from the alternatives. The authors proceed from lesser rigor to greater within each chapter, presenting definitions, theorems, and worked examples before the proofs, which are placed at the end of each chapter. The authors address this somewhat unusual organization in their introduction:"We decided to retain the format of the first edition, which gives full technical proofs at the end of each chapter but presents some idea of the main point in the text. This seems to have been well-received by the majority of readers... and we still believe that it is a sound pedagogical device for a course like this. It is not meant as a way to shun the proofs; on the contrary it is intended to give to views of the proof: on in the way working mathematicians think about it, (the trade secrets, so to speak), and the other in the way mathematicians write out formal proofs." Marsden Hoffman is written in a slightly more conversational tone than other rigorous introductions to analysis. However, as a math major at Stanford, I felt like this only made the text more readable.A side note: Though Marsden and Hoffman do make light of Cantor's quaint, 19th century definition of a set in their intro to set theory, they ultimately do so only to motivate the exposition of a formal, axiomatic view.