Introduction to Analysis [Lecture notes]

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Evan Chen • Course Notes

The one-dimensional case is discussed simultaneously with, and as a special case of, the n -dimensional case. This is true as well for the final chapter on differential forms. There are, however, plenty of interesting and challenging Group I problems as well. Sometimes, well-known results such as the completeness of a compact metric space, or the fact that a nonempty perfect subset of a complete metric space is uncountable appear as Group II exercises.

No solutions appear in the text and there is, to my knowledge, no solutions manual provided by the publisher. One other thing that is omitted is a bibliography or list of suggestions for further reading.

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This seems unfortunate: I think it is never too early to introduce mathematics majors to the idea of using a library and glancing at other books on a subject. I think that this is a classic example of one of those texts that is likely to appeal more to faculty members than to beginning students or at least the kind of beginning students that one would find at a place like Iowa State University. It is well-written, but assumes a level of mathematical sophistication on the part of the reader that the vast majority of undergraduate students simply do not have.

To really appreciate the concise elegance of the exposition, considerable prior understanding of this material is a must.

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I enjoyed the course and the book, but found the latter to be frequently difficult reading. Rudin, for all his conciseness, did not do that: differentiation was first defined for real-valued functions of a real variable, for example, and only after that concept was assimilated did he progress to the multivariable situation.

The bottom line, therefore, is that while this is an interesting and well-written book, it is, notwithstanding its title, not appropriate as a text for a typical introductory analysis course at a typical university.

However, for very sophisticated honors courses with exceptionally well-prepared students who are prepared to work hard, it would certainly offer a vivid, exciting and informative introduction to this material. Mark Hunacek mhunacek iastate.

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