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Problems and Solutions for Undergraduate Analysis : Undergraduate Texts in Mathematics - Rami Shakarchi

Problems and Solutions for Undergraduate Analysis

Undergraduate Texts in Mathematics


Published: 19th December 1997
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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, ยง5, 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.

Sets and Mappingsp. 1
Real Numbersp. 9
Limits and Continuous Functionsp. 19
Differentiationp. 35
Elementary Functionsp. 43
The Elementary Real Integralp. 73
Normed Vector Spacesp. 91
Limitsp. 111
Compactnessp. 125
Seriesp. 133
The Integral in One Variablep. 165
Approximation with Convolutionsp. 183
Fourier Seriesp. 189
Improper Integralsp. 217
The Fourier Integralp. 243
Functions on n-Spacep. 253
The Winding Number and Global Potential Functionsp. 287
Derivatives in Vector Spacesp. 293
Inverse Mapping Theoremp. 303
Ordinary Differential Equationsp. 327
Multiple Integralsp. 337
Differential Formsp. 359
Table of Contents provided by Blackwell. All Rights Reserved.

ISBN: 9780387982359
ISBN-10: 0387982353
Series: Undergraduate Texts in Mathematics
Audience: General
Format: Paperback
Language: English
Number Of Pages: 368
Published: 19th December 1997
Publisher: Springer-Verlag New York Inc.
Country of Publication: US
Dimensions (cm): 23.9 x 15.8  x 2.3
Weight (kg): 0.62