Spaces is a modern introduction to real analysis at the advanced undergraduate level. It is forward-looking in the sense that it first and foremost aims to provide students with the concepts and techniques they need in order to follow more advanced courses in mathematical analysis and neighboring fields. The only prerequisites are a solid understanding of calculus and linear algebra. Two introductory chapters will help students with the transition from computation-based calculus to theory-based analysis. The main topics covered are metric spaces, spaces of continuous functions, normed spaces, differentiation in normed spaces, measure and integration theory, and Fourier series. Although some of the topics are more advanced than what is usually found in books of this level, care is taken to present the material in a way that is suitable for the intended audience: concepts are carefully introduced and motivated, and proofs are presented in full detail. Applications to differential equations and Fourier analysis are used to illustrate the power of the theory, and exercises of all levels from routine to real challenges help students develop their skills and understanding. The text has been tested in classes at the University of Oslo over a number of years.

Table of Contents

Chapter 1. Preliminaries: Proofs, Sets, and Functions

Chapter 2. The Foundation of Calculus

Chapter 3. Metric Spaces

Chapter 4. Spaces of Continuous Functions

Chapter 5. Normed Spaces and Linear Operators

Chapter 6. Differential Calculus in Normed Spaces

Chapter 7. Measure and Integration

Chapter 8. Constructing Measures

Chapter 9. Fourier Series

Review

[T]he presentation is done in a way to make the book eminently readable by undergraduate students...I think that reading 'Spaces' or taking a course based on the text would serve very well as a bridge between undergraduate level and modern graduate level mathematics. --Jason M. Graham, MAA Reviews

About the Author

Tom L. Lindstrom, University of Oslo, Norway.