# Problems and Solutions for Undergraduate Analysis

• Rami Shakarchi
Textbook

1. Front Matter
Pages i-xii
2. Rami Shakarchi
Pages 1-8
3. Rami Shakarchi
Pages 9-17
4. Rami Shakarchi
Pages 19-34
5. Rami Shakarchi
Pages 35-41
6. Rami Shakarchi
Pages 43-72
7. Rami Shakarchi
Pages 73-89
8. Rami Shakarchi
Pages 91-110
9. Rami Shakarchi
Pages 111-124
10. Rami Shakarchi
Pages 125-131
11. Rami Shakarchi
Pages 133-164
12. Rami Shakarchi
Pages 165-182
13. Rami Shakarchi
Pages 183-187
14. Rami Shakarchi
Pages 189-215
15. Rami Shakarchi
Pages 217-241
16. Rami Shakarchi
Pages 243-251
17. Rami Shakarchi
Pages 253-285
18. Rami Shakarchi
Pages 287-291
19. Rami Shakarchi
Pages 293-302
20. Rami Shakarchi
Pages 303-326
21. Rami Shakarchi
Pages 327-335
22. Rami Shakarchi
Pages 337-358
23. Rami Shakarchi
Pages 359-368

### Introduction

The present volume contains all the exercises and their solutions for Lang's second edition of Undergraduate Analysis. The wide variety of exercises, which range from computational to more conceptual and which are of vary­ ing difficulty, cover the following subjects and more: real numbers, limits, continuous functions, differentiation and elementary integration, normed vector spaces, compactness, series, integration in one variable, improper integrals, convolutions, Fourier series and the Fourier integral, functions in n-space, derivatives in vector spaces, the inverse and implicit mapping theorem, ordinary differential equations, multiple integrals, and differential forms. My objective is to offer those learning and teaching analysis at the undergraduate level a large number of completed exercises and I hope that this book, which contains over 600 exercises covering the topics mentioned above, will achieve my goal. The exercises are an integral part of Lang's book and I encourage the reader to work through all of them. In some cases, the problems in the beginning chapters are used in later ones, for example, in Chapter IV when one constructs-bump functions, which are used to smooth out singulari­ ties, and prove that the space of functions is dense in the space of regu­ lated maps. The numbering of the problems is as follows. Exercise IX. 5. 7 indicates Exercise 7, §5, of Chapter IX. Acknowledgments I am grateful to Serge Lang for his help and enthusiasm in this project, as well as for teaching me mathematics (and much more) with so much generosity and patience.

### Keywords

Derivative Fourier series calculus compactness convolution differential equation

#### Authors and affiliations

• Rami Shakarchi
• 1
1. 1.Department of MathematicsPrinceton UniversityPrincetonUSA

### Bibliographic information

• DOI https://doi.org/10.1007/978-1-4612-1738-1
• Copyright Information Springer-Verlag New York, Inc. 1998
• Publisher Name Springer, New York, NY
• eBook Packages
• Print ISBN 978-0-387-98235-9
• Online ISBN 978-1-4612-1738-1
• Buy this book on publisher's site