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# Problems and Solutions for Undergraduate Analysis

Paperback Published: 19th December 1997
ISBN: 9780387982359
Number Of Pages: 368

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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, Â§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.

 Preface Sets and Mappings p. 1 Real Numbers p. 9 Limits and Continuous Functions p. 19 Differentiation p. 35 Elementary Functions p. 43 The Elementary Real Integral p. 73 Normed Vector Spaces p. 91 Limits p. 111 Compactness p. 125 Series p. 133 The Integral in One Variable p. 165 Approximation with Convolutions p. 183 Fourier Series p. 189 Improper Integrals p. 217 The Fourier Integral p. 243 Functions on n-Space p. 253 The Winding Number and Global Potential Functions p. 287 Derivatives in Vector Spaces p. 293 Inverse Mapping Theorem p. 303 Ordinary Differential Equations p. 327 Multiple Integrals p. 337 Differential Forms p. 359 Table of Contents provided by Blackwell. All Rights Reserved.

ISBN: 9780387982359
ISBN-10: 0387982353