The fourth edition incorporates a large number of corrections reported since the release of the third edition, as well as some new exercises. 2 TERENCE TAO for the limiting case when one or both of the pi are equal to innity we leave this to the reader.) Similarly we can factorise g G 1 0 G 1b where G0,G1 are non-negative simple functions with Lq 0(Y ) and Lq 1(Y ) norms respectively equal to 1, and b is a simple function of magnitude at most 1. 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 twenty-five to thirty lectures each. Moreover, original exercises, while better motivated, are too few and too easy in comparison. There are also appendices on mathematical logic and the decimal system. Tao is quite frankly not even on par: granted it is more modern and it contains all you will need in a mathematical analysis 1 course, it is too easy, and proofs suck (they are morbid: 1.5 pages proofs of trivial facts because of extreme verbosity). 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. These are my notes on the books Analysis I and Analysis II, 4th edition, written by Terence Tao.My notes cover statements I found important or interesting, including almost all axioms, corollaries, exercises, lemmata, proofs, propositions, remarks, and theories in both books. The material starts at the very beginning-the construction of the 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. The emphasis is on rigour and on foundations. 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.
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