Analysis II 3rd Edition by Terence Tao 9811018049 9789811018046

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ISBN-10 : 9811018049 

ISBN-13 : 9789811018046

Author: Terence Tao

This 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.   

 

Analysis II 3rd Table of contents:

1 Metric spaces

1.1 Definitions and examples

1.2 Some point-set topology of metric spaces

1.3 Relative topology

1.4 Cauchy sequences and complete metric spaces

1.5 Compact metric spaces

2 Continuous functions on metric spaces

2.1 Continuous functions

2.2 Continuity and product spaces

2.3 Continuity and compactness

2.4 Continuity and connectedness

2.5 Topological spaces (Optional)

3 Uniform convergence

3.1 Limiting values of functions

3.2 Pointwise and uniform convergence

3.3 Uniform convergence and continuity

3.4 The metric of uniform convergence

3.5 Series of functions; the Weierstrass M-test

3.6 Uniform convergence and integration

3.7 Uniform convergence and derivatives

3.8 Uniform approximation by polynomials

4 Power series

4.1 Formal power series

4.2 Real analytic functions

4.3 Abel’s theorem

4.4 Multiplication of power series

4.5 The exponential and logarithm functions

4.6 A digression on complex numbers

4.7 Trigonometric functions

5 Fourier series

5.1 Periodic functions

5.2 Inner products on periodic functions

5.3 Trigonometric polynomials

5.4 Periodic convolutions

5.5 The Fourier and Plancherel theorems

6 Several variable differential calculus

6.1 Linear transformations

6.2 Derivatives in several variable calculus

6.3 Partial and directional derivatives

6.4 The several variable calculus chain rule

6.5 Double derivatives and Clairaut’s theorem

6.6 The contraction mapping theorem

6.7 The inverse function theorem in several variable calculus

6.8 The implicit function theorem

7 Lebesgue measure

7.1 The goal: Lebesgue measure

7.2 First attempt: Outer measure

7.3 Outer measure is not additive

7.4 Measurable sets

7.5 Measurable functions

8 Lebesgue integration

8.1 Simple functions

8.2 Integration of non-negative measurable functions

8.3 Integration of absolutely integrable functions

8.4 Comparison with the Riemann integral

8.5 Fubini’s theorem

 

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