An Introduction to Real Analysis

✍ Scribed by John K. Hunter

• Year: 2014
• Tongue: English
• Leaves: 305
• Category: Library

✦ Synopsis

These are some notes on introductory real analysis. They cover the properties of the real numbers, sequences and series of real numbers, limits of functions, continuity, differentiability, sequences and series of functions, and Riemann integration. They don’t include multi-variable calculus or contain any problem sets. Optional sections are starred.

✦ Table of Contents

Chapter 1. Sets and Functions
1.1. Sets
1.2. Functions
1.3. Composition and inverses of functions
1.4. Indexed sets
1.5. Relations
1.6. Countable and uncountable sets
Chapter 2. Numbers
2.1. Integers
2.2. Rational numbers
2.3. Real numbers: algebraic properties
2.4. Real numbers: ordering properties
2.5. The supremum and infimum
2.6. Real numbers: completeness
2.7. Properties of the supremum and infimum
Chapter 3. Sequences
3.1. The absolute value
3.2. Sequences
3.3. Convergence and limits
3.4. Properties of limits
3.5. Monotone sequences
3.6. The limsup and liminf
3.7. Cauchy sequences
3.8. Subsequences
3.9. The Bolzano-Weierstrass theorem
Chapter 4. Series
4.1. Convergence of series
4.2. The Cauchy condition
4.3. Absolutely convergent series
4.4. The comparison test
4.5. * The Riemann -function
4.6. The ratio and root tests
4.7. Alternating series
4.8. Rearrangements
4.9. The Cauchy product
4.10. * Double series
4.11. * The irrationality of e
Chapter 5. Topology of the Real Numbers
5.1. Open sets
5.2. Closed sets
5.3. Compact sets
5.4. Connected sets
5.5. * The Cantor set
Chapter 6. Limits of Functions
6.1. Limits
6.2. Left, right, and infinite limits
6.3. Properties of limits
Chapter 7. Continuous Functions
7.1. Continuity
7.2. Properties of continuous functions
7.3. Uniform continuity
7.4. Continuous functions and open sets
7.5. Continuous functions on compact sets
7.6. The intermediate value theorem
7.7. Monotonic functions
Chapter 8. Differentiable Functions
8.1. The derivative
8.2. Properties of the derivative
8.3. The chain rule
8.4. Extreme values
8.5. The mean value theorem
8.6. Taylor's theorem
8.7. * The inverse function theorem
8.8. * L'Hôspital's rule
Chapter 9. Sequences and Series of Functions
9.1. Pointwise convergence
9.2. Uniform convergence
9.3. Cauchy condition for uniform convergence
9.4. Properties of uniform convergence
9.5. Series
Chapter 10. Power Series
10.1. Introduction
10.2. Radius of convergence
10.3. Examples of power series
10.4. Algebraic operations on power series
10.5. Differentiation of power series
10.6. The exponential function
10.7. * Smooth versus analytic functions
Chapter 11. The Riemann Integral
11.1. The supremum and infimum of functions
11.2. Definition of the integral
11.3. The Cauchy criterion for integrability
11.4. Continuous and monotonic functions
11.5. Linearity, monotonicity, and additivity
11.6. Further existence results
11.7. * Riemann sums
11.8. * The Lebesgue criterion
Chapter 12. Properties and Applications of the Integral
12.1. The fundamental theorem of calculus
12.2. Consequences of the fundamental theorem
12.3. Integrals and sequences of functions
12.4. Improper Riemann integrals
12.5. * Principal value integrals
12.6. The integral test for series
12.7. Taylor's theorem with integral remainder
Chapter 13. Metric, Normed, and Topological Spaces
13.1. Metric spaces
13.2. Normed spaces
13.3. Open and closed sets
13.4. Completeness, compactness, and continuity
13.5. Topological spaces
13.6. * Function spaces
13.7. * The Minkowski inequality
Bibliography