Analysis II

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

• Index
• Texts and Readings in Mathematics