Truly excellent book, 338 pages of mathematical TNT.

If I could only keep one of my maths library, this would be it.

That's not to say I'm not going to have a go at rewriting some of his proofs as I think they could be a bit clearer (in particular 5.15 Taylor's Theorem, which also needs correct attribution to James Gregory).

For the benefit of @Floyd3345 below, mine is the International 3rd edition (1976) as per the cover photo at left, if you want to check any text in particular ( )

Made me hate myself but I came out stronger in the end. ( )

The copy of Principles of Mathematical Analysis by Walter Rudin that I own is interesting in one way; it states that it is the Indian Edition. Now I don’t know much about publishing, but the biggest issue for me was whether or not the book was in English since I don’t know any Indian languages. I mean, I suppose the paper making up the book is slightly thinner and perhaps it uses a different measure of size, but other than that it didn’t need to say that on the cover.

This mathematical book is much like any other mathematical textbook that I own and have read, it starts with the basics and builds upon those basics in a systematic manner. The book contains proofs of theorems and practice problems, making it a very good resource.

I have heard that this book is used as a textbook in classes, but I never had to take a class in Analysis. As I might have mentioned long ago, all of the books that I read are only for my own amusement. However, it would be neat if I also learned something along the way.

The book delivers in being amusing and informative. I suppose it might be less amusing if this was a book I was assigned, but that is beside the point. In being informative, the book contains eleven chapters and covers subjects from The Real and Complex Number Systems to Lebesgue Theory. Finally, the book has a bibliography, an index, and a list of the special symbols used in the book. ( )

Rudin is exact, specific, and direct. This is a harsh and austere text, but allows for very meditative readings. ( )

Indeholder "Preface", "Chapter 1: The Real and Complex Number Systems", " Introduction", " Dedekind cuts", " Real numbers", " The extended real number system", " Complex numbers", " Exercises", "Chapter 2: Elements of Set Theory", " Finite, countable, and uncountable sets", " Metric spaces", " Compact sets", " Perfect sets", " Exercises", "Chapter 3: Numerical Sequences and Series", " Fundamental properties of sequences", " Subsequences", " The Cauchy criterion", " Upper and lower limits", " Some special sequences", " Series", " Series of nonnegative terms", " The number e", " The root and ratio tests", " Power series", " Partial summation", " Absolute convergence", " Addition and multiplication of series", " Rearrangements", " Exercises", "Chapter 4: Continuity", " Introduction", " The limit of a function", " Continuous functions", " Continuity and compactness", " Discontinuities", " The Bolzano-Weierstrass theorem", " Monotonic functions", " Infinite limits and limits at infinity", " Exercises", "Chapter 5: Differentiation", " The derivative of a function", " The mean value theorems", " The continuity of derivatives", " L'Hospital's rule", " Derivatives of higher order", " Taylor's theorem", " Exercises", "Chapter 6: The Riemann-Stieltjes Integral", " Definition and existence of the integral", " The integral as a limit of sums", " The scope of Stieltjes's integration", " The fundamental theorem of calculus", " Functions of bounded variation", " Generalization of the integral", " Some further theorems on integration", " Curves, and the length of a curve", " Exercises", "Chapter 7: Sequences and Series of Functions", " Discussion of main problem", " Uniform convergence", " Uniform convergence and continuity", " Uniform convergence and integration", " Uniform convergence and differentiation", " Equicontinuous families of functions", " The Stone-Weierstrass theorem", " Exercises", "Chapter 8: Further Topics in the Theory of Series", " Power series", " The exponential and logarithmic functions", " The trigonometric functions", " Fourier series", " Exercises", "Chapter 9: Functions of Several Variables", " Continuous transformations of metric spaces", " Euclidean spaces", " Continuous functions of several real variables", " Partial derivatives", " Linear transformations and determinants", " The inverse function theorem", " The implicit function theorem", " Functional dependence", " Exercises", "Chapter 10: The Lebesgue Theory", " Set functions", " Construction of the Lebesgue measure", " Measure spaces", " Measurable functions", " Simple functions", " Integration", " Comparison with the Riemann integral", " Functions of Class L^2", " Exercises", "Bibliography", "List of Frequently Occurring Symbols and Abbreviations", "Index".

En superklassiker indenfor feltet og også den første, så den er nærmest kult. "It was easy to say, and often true, that anyone who could survive a year of Rudin was a real mathematician."
Bogen er ret hård kost uden ret mange eksempler, beviser som mest er hints og skitser, og så en stak øvelser, som er benhårde. ( )