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Analysis I (Texts and Readings in Mathematics) [Englisch] [Gebundene Ausgabe]

Terence Tao

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15. Januar 2009 Texts and Readings in Mathematics
This is part one of a two-volume introduction to real analysis and is intended for honours undergraduates, who have already been exposed to calculus. The emphasis is on rigour and on foundations. The material starts at the very beginning - the construction of number systems and set theory, then goes on to the basics of analysis (limits, series, continuity, differentiation, Riemann integration), through to power series, several variable calculus and Fourier analysis, and finally to the Lebesgue integral. These are almost entirely set in the concrete setting of the real line and Euclidean spaces, although there is some material on abstract metric and topological spaces. There are appendices on mathematical logic and the decimal system. The entire text (omitting some less central topics) can be taught in two quarters of twenty-five to thirty lectures each. The course material is deeply intertwined with the exercises, as it is intended that the student actively learn the material (and practice thinking and writing rigorously) by proving several of the key results in the theory. The second edition has been extensively revised and updated.

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Die hilfreichsten Kundenrezensionen auf (beta) 5.0 von 5 Sternen  1 Rezension
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5.0 von 5 Sternen Excellent introduction (review) to the theory of calculus 30. Dezember 2013
Von Anonymous statman - Veröffentlicht auf
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Real analysis courses are one of the most challenging topics for students. Most books basically go into the nitty-gritty of the subject right away. But it is challenging for students to abandon what they have already been thought about number systems and mathematical operations to rigorously learn about their meaning. Perhaps not surprisingly, it took a gifted mathematician (who got his PhD at age 20 or so) to come up with the ultimate real analysis textbook.

In Analysis I, Tao introduces rigorous development of analysis concepts in a clear and transparent way. Literally all bases are covered, starting by covering more elementary material that usually is not discussed in real analysis courses. Even that the parenthesis is used in different contexts (e.g. to define functions and coordinates) is remarked. The author proves very clearly each mathematical statement at first, setting the tone for the reader to think critically. Little by little more and more proofs are left to the reader, acknowledging the need of doing exercises to learn the material. Many of the exercises provides useful Hints though. By analysis II virtually all theorem, lemma, corollary and proposition proofs are left to the reader. Some may find the lack of proofs (mostly in Analysis II) challenging, but many of these are discussed in class anyway, or can be found in other real analysis textbooks if need be. Analysis II is still a very useful volume due to the explanations leading to the mathematical statements and its chapters on Lebesgue measures and Lebesgue intregration.

The books are excellent for use in an undergraduate course or for those in need of a review on the subject (such as myself). The authors blog includes additional help on this and other math topics.

I do think the quality of the book material can be better though. The print sometimes is opaque and the binding is weak. In terms of content, there's room for improvement on the index. But overall, this is an incredibly useful resource for those interested in mathematical and probability theory.
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