- Volume
-
1 ISBN 9789380250649
Description
This is part one of a two-volume introduction to real analysis and is intended for honours undergraduates who have already been exposed to calculus. The emphasis is on rigour and on foundations. The material starts at the very beginning - the construction of the number systems and set theory - then goes on to the basics of analysis (limits, series, continuity, differentiation, Riemann integration), through to power series, several variable calculus and Fourier analysis, and finally to 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. There are also appendices on mathematical logic and the decimal system. The entire text (omitting some less central topics) can be taught in two quarters of twenty-five to thirty 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.
In the third edition, several typos and other errors have been corrected and a few new exercises have been added.
Table of Contents
1. Introduction
2. Starting at the beginning: the natural numbers
3. Set theory
4. Integers and rationals
5. The real numbers
6. Limits of sequences
7. Series
8. Infinite sets
9. Continuous functions on R
10. Differentiation of functions
11. The Riemann integral
Appendix A: The basics of mathematical logic
Appendix B: The decimal system
- Volume
-
2 ISBN 9789380250656
Description
This is part two of a two-volume introduction to real analysis and is intended for honours undergraduates who have already been exposed to calculus. The emphasis is on rigour and on foundations. The material starts at the very beginning - the construction of the number systems and set theory - then goes on to the basics of analysis (limits, series, continuity, differentiation, Riemann integration), through to power series, several variable calculus and Fourier analysis, and finally to 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. There are also appendices on mathematical logic and the decimal system. The entire text (omitting some less central topics) can be taught in two quarters of twenty-five to thirty 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.
In the third edition, several typos and other errors have been corrected and a few new exercises have been added.
Table of Contents
1. Metric spaces
2. Continuous functions on metric spaces
3. Uniform convergence
4. Power series
5. Fourier series
6. Several variable differential calculus
7. Lebesgue measure
8. Lebesgue integration
by "Nielsen BookData"