Methods of Real Analysis - ISBN:0471310654

Methods of Real Analysis

Date: 21 April 2011, 14:17

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Summary: Great Text with Great Proofs
Rating: 5
In some presentations rigor is sacrificed for an intuitive overview of the material. This is a mistake in my view, particularly for an undergraduate course in analysis, which is the perfect environment for rigorous proof-writing skills to be developed.
Goldberg makes no such sacrifice. His proofs are great. His exposition is very clear. If I have a complaint, it's that it's not colorful enough, there aren't enough different kinds of pictures and graphs. The text DOES contain pictures and diagrams, and they are good, but they are not great. On the other hand, this might be seen as a feature, for it forces the student to work on coming up with her own, which is a very important skill to develop.
Summary: Great introduction to Real Analysis
Rating: 5
This book reads like an instructor would teach in class. It derives all the important theorems quite rigorously and throws in a few lines of intuition which is very helpful when you are trying to self-study something as intense as real analysis. He also has two or three examples following every major result and shows clearly how to "use" the result just derived to solve an actual math problem.
I went through lots of great analysis books (Rudin, Shilov, Kolmogorov, Aliprantis, Johnsonbaugh, Rosenlicht and Protter among others) until finally learning from this. After getting my foundations and intuitions right, I now feel like I am better equipped to read and understand the results from the above books which, in general, treat proofs more tersely and cover a lot more material in the process.
This book does not cover the origin of real numbers and it's axioms. It covers standard results from elementary set theory, sequences, limits, metric spaces, open and closed sets, completeness, compactness and connectedness, the derivative, Taylor Series, exps and logs, the Lebesgue integral and Fourier Series.

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