Analysis I 4th Edition by Terence Tao ISBN 9789811972614 9811972613

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ISBN 10: 9811972613
ISBN 13: 9789811972614
Author: Terence Tao

Analysis I 4th Edition Table of contents:

1 Introduction

1.1 What Is Analysis?

1.2 Why Do Analysis?

2 Starting at the Beginning: The Natural Numbers

2.1 The Peano Axioms

2.2 Addition

2.3 Multiplication

3 Set Theory

3.1 Fundamentals

3.2 Russell’s Paradox (Optional)

3.3 Functions

3.4 Images and Inverse Images

3.5 Cartesian Products

3.6 Cardinality of Sets

4 Integers and Rationals

4.1 The Integers

4.2 The Rationals

4.3 Absolute Value and Exponentiation

4.4 Gaps in the Rational Numbers

5 The Real Numbers

5.1 Cauchy Sequences

5.2 Equivalent Cauchy Sequences

5.3 The Construction of the Real Numbers

5.4 Ordering the Reals

5.5 The Least Upper Bound Property

5.6 Real Exponentiation, Part I

6 Limits of Sequences

6.1 Convergence and Limit Laws

6.2 The Extended Real Number System

6.3 Suprema and Infima of Sequences

6.4 Limsup, Liminf, and Limit Points

6.5 Some Standard Limits

6.6 Subsequences

6.7 Real Exponentiation, Part II

7 Series

7.1 Finite Series

7.2 Infinite Series

7.3 Sums of Non-negative Numbers

7.4 Rearrangement of Series

7.5 The Root and Ratio Tests

8 Infinite Sets

8.1 Countability

8.2 Summation on Infinite Sets

8.3 Uncountable Sets

8.4 The Axiom of Choice

8.5 Ordered Sets

9 Continuous Functions on R

9.1 Subsets of the Real Line

9.2 The Algebra of Real-Valued Functions

9.3 Limiting Values of Functions

9.4 Continuous Functions

9.5 Left and Right Limits

9.6 The Maximum Principle

9.7 The Intermediate Value Theorem

9.8 Monotonic Functions

9.9 Uniform Continuity

9.10 Limits at Infinity

10 Differentiation of Functions

10.1 Basic Definitions

10.2 Local Maxima, Local Minima, and Derivatives

10.3 Monotone Functions and Derivatives

10.4 Inverse Functions and Derivatives

10.5 L’Hôpital’s Rule

11 The Riemann Integral

11.1 Partitions

11.2 Piecewise Constant Functions

11.3 Upper and Lower Riemann Integrals

11.4 Basic Properties of the Riemann Integral

11.5 Riemann Integrability of Continuous Functions

11.6 Riemann Integrability of Monotone Functions

11.7 A Non-riemann Integrable Function

11.8 The Riemann–Stieltjes Integral

11.9 The Two Fundamental Theorems of Calculus

11.10 Consequences of the Fundamental Theorems

Appendix A The Basics of Mathematical Logic

A.1 Mathematical Statements

A.2 Implication

A.3 The Structure of Proofs

A.4 Variables and Quantifiers

A.5 Nested Quantifiers

A.6 Some Examples of Proofs and Quantifiers

A.7 Equality

Appendix B The Decimal System

B.1 The Decimal Representation of Natural Numbers

B.2 The Decimal Representation of Real Numbers

Index

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