This book is an outline of the core material in the standard graduate-level real analysis course. It is intended as a resource for students in such a course as well as others who wish to learn or review the subject. On the abstract level, it covers the theory of measure and integration and the basics of point set topology, functional analysis, and the most important types of function spaces. On the more concrete level, it also deals with the applications of these general theories to analysis on Euclidean space: the Lebesgue integral, Hausdorff measure, convolutions, Fourier series and transforms, and distributions. The relevant definitions and major theorems are stated in detail. Proofs, however, are generally presented only as sketches, in such a way that the key ideas are explained but the technical details are omitted. In this way a large amount of material is presented in a concise and readable form.
Prologue: Notation, Terminology, and Set Theory
2. Measure and Integration: General Theory
3. Measure and Integration
4. Rudiments of Functional Analysis
6. Topics in Analysis on Euclidean Space
Gerald B. Folland was born and raised in Salt Lake City, Utah. He received his bachelor’s degree from Harvard University in 1968 and his doctorate from Princeton University in 1971. After two years at the Courant Institute, he moved to the University of Washington, where he is now professor of mathematics. He is the author of ten textbooks and research monographs in the areas of real analysis, harmonic analysis, partial differential equations, and mathematical physics.
This text is an entry in the MAA Guides series, each volume of which is intended to provide a short, concentrated summary of a primary mathematical subject. This volume on real analysis focuses on the material in a standard first graduate course. It is ideally suited to serve as a quick look for someone new to the subject, for review, or as preparation for qualifying exams. It is concise, very clearly written, and full of little nuggets of insight. Continued...