Tao provides excellent treatment of sequences, series, and continuity. Motivating the study of analysis via the construction of the number systems vis vi peano arithmetic, provides the reader with a road map for properly conducting mathematical reasoning in highly abstract and proof based settings. Taos unique ability to combine his other worldly mathematical skill set with writing that is crisp and engaging. Terrence Tao has published an incredibly comprehensive textbook that serves as a wonderful introduction to the subject for interested outsiders and undergrad students in mathematics. 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 entire text (omitting some less central topics) can be taught in two quarters of 25–30 lectures each. The book also has appendices on mathematical logic and the decimal system. 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. Beginning with the construction of the number systems and set theory, the book discusses the basics of analysis (limits, series, continuity, differentiation, Riemann integration), through to power series, several variable calculus and Fourier analysis, and then finally the Lebesgue integral. The emphasis is on rigour and foundations of analysis. ![]() ![]() This is part one of a two-volume book on real analysis and is intended for senior undergraduate students of mathematics who have already been exposed to calculus.
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