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 varyยญ 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 singulariยญ ties, and prove that the space of functions is dense in the space of reguยญ 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
CONTENT
0 Sets and Mappings -- I Real Numbers -- II Limits and Continuous Functions -- III Differentiation -- IV Elementary Functions -- V The Elementary Real Integral -- VI Normed Vector Spaces -- VII Limits -- VIII Compactness -- IX Series -- X The Integral in One Variable -- XI Approximation with Convolutions -- XII Fourier Series -- XIII Improper Integrals -- XIV The Fourier Integral -- XV Functions on n-Space -- XVI The Winding Number and Global Potential Functions -- XVII Derivatives in Vector Spaces -- XVIII Inverse Mapping Theorem -- XIX Ordinary Differential Equations -- XX Multiple Integrals -- XXI Differential Forms
