LECTURE NOTES OF WILLIAM CHEN
FUNDAMENTALS OF ANALYSIS
This set of notes has been organized in such a way to create a single volume suitable for an introduction to some of the basic ideas in analysis. Chapters 1 - 4 and 7 - 8 were used in various forms and on many occasions between 1982 and 1990 by the author at Imperial College, University of London. Chapters 5 and 6 were added in Sydney in 1994 and 1996 respectively.
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Chapter 1 : THE NUMBER SYSTEM >>
- The Real Numbers
- Completeness of the Real Numbers
- The Complex Numbers
- Countability
- Cardinal Numbers
Chapter 2 : SEQUENCES AND LIMITS >>
- Introduction
- Real Sequences
- Tests for Convergence
- Recurrence Relations
- Subsequences
Chapter 3 : SERIES >>
- Introduction
- Real Series
- Complex Series
- Power Series
- Multiplication of Series
Chapter 4 : FUNCTIONS AND CONTINUITY >>
- Limits of Functions
- Continuity in Intervals
- Continuity in Closed Intervals
Chapter 5 : DIFFERENTIATION >>
- Introduction
- Some Important Results on Derivatives
- Stationary Points and Second Derivatives
- Series Expansion
Chapter 6 : THE RIEMANN INTEGRAL >>
- Introduction
- Properties of the Riemann Integral
- Sufficient Conditions for Integrability
- Integration as the Inverse of Differentiation
- An Important Example
Chapter 7 : FURTHER TREATMENT OF LIMITS >>
- Upper and Lower Limits of a Real Sequence
- Double and Repeated Limits
- Infinite Products
- Double Integrals
Chapter 8 : UNIFORM CONVERGENCE >>
- Introduction
- Criteria for Uniform Convergence
- Consequences of Uniform Convergence
- Application to Power Series