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Problems and Solutions for Undergraduate Analysis

  • Rami Shakarchi

Table of contents

  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 293-302
  19. Rami Shakarchi
    Pages 303-326
  20. Rami Shakarchi
    Pages 327-335
  21. Rami Shakarchi
    Pages 337-358
  22. Rami Shakarchi
    Pages 359-368

About this book

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 Springer Book Archive
  • Print ISBN 978-0-387-98235-9
  • Online ISBN 978-1-4612-1738-1
  • Buy this book on publisher's site
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