Tao T. Analysis II

3rd Edition. — Springer Science+Business Media, Singapore, 2016. — 235 p. — (Texts and Readings in Mathematics. Volume 38). — ISBN: 9811018030This is part two of a two-volume book on real analysis and is intended for senior undergraduate students of mathematics who have already been exposed to calculus. The emphasis is on rigour and foundations of analysis. 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. 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. The book also has appendices on mathematical logic and the decimal system. The entire text (omitting some less central topics) can be taught in two quarters of 25–30 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.Metric spaces
Continuous functions on metric spaces
Uniform convergence
Power series
Fourier series
Several variable differential calculus
Lebesgue measure
Lebesgue integration