Analysis I

his 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.

The fourth edition incorporates a large number of additional corrections reported since the release of the third edition, as well as some additional new exercises.

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

About the Author

Terence Tao is an Australian and American mathematician who is a professor of mathematics at the University of California, Los Angeles, where he holds the James and Carol Collins Chair in the College of Letters and Sciences.