Synopsis:
An essential undergraduate textbook on algebra, topology, and calculus An Introduction to Analysis is an essential primer on basic results in algebra, topology, and calculus for undergraduate students considering advanced degrees in mathematics. Ideal for use in a one-year course, this unique textbook also introduces students to rigorous proofs and formal mathematical writing--skills they need to excel. With a range of problems throughout, An Introduction to Analysis treats n-dimensional calculus from the beginning--differentiation, the Riemann integral, series, and differential forms and Stokes's theorem--enabling students who are serious about mathematics to progress quickly to more challenging topics. The book discusses basic material on point set topology, such as normed and metric spaces, topological spaces, compact sets, and the Baire category theorem. It covers linear algebra as well, including vector spaces, linear mappings, Jordan normal form, bilinear mappings, and normal mappings. Proven in the classroom, An Introduction to Analysis is the first textbook to bring these topics together in one easy-to-use and comprehensive volume. Provides a rigorous introduction to calculus in one and several variables Introduces students to basic topology Covers topics in linear algebra, including matrices, determinants, Jordan normal form, and bilinear and normal mappings Discusses differential forms and Stokes's theorem in n dimensions Also covers the Riemann integral, integrability, improper integrals, and series expansions
Review:
"Gunning's book is a great introduction to analysis that presents precisely what an honors analysis course should include. The writing is rigorous but lively, and much interesting mathematics is packed in it."--Wilhelm Schlag, University of Chicago
"A valuable, wonderful book."--Camil Muscalu, Cornell University
"An excellent and unusual book by an esteemed mathematician and teacher. An Introduction to Analysis is a uniquely valuable sourcebook for much of undergraduate mathematics."--John B. Garnett, University of California, Los Angeles
"One would expect that this book offers a demanding, sophisticated look at this material, done rapidly and concisely. One would be correct. . . . [F]or very sophisticated honors courses with exceptionally well-prepared students who are prepared to work hard, it would certainly offer a vivid, exciting and informative introduction to this material."---Mark Hunacek, MAA Reviews
"The ideal textbook to cover the foundations of mathematics. . . . This is one of the best books at the undergraduate level that I've ever read."---Jonathan Shock, Mathemafrica
"The book is clearly written, with elegant proofs, and a clever selection of the topics, allowing to present in one volume in a coherent and self-contained manner, several topics, starting with background results in abstract algebra and topology and culminating with Stokes' theorem. . . . The book will be of great help for teachers on advanced courses in analysis, or for self-study by students, giving them a quick but rigorous acquittance with somefundamental results of mathematical analysis."---Tiberiu Trif, Studia Mathematica
"Tested and proven in classroom, this unique undergraduate textbook presents various basic topics in one coherent primer of reasonable size, and that in a very elegant style."---Werner Kleinert, Zentralblatt MATH
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