A Guide to Advanced Real Analysis 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.

This work serves as a Baedeker to the more accessible parts of the terrain of advanced real analysis, providing an overview for those less familiar, a refresher for those more so, and a key to the features of that terrain, including what high points and cultural monuments to visit if planning to explore the subject more seriously. Intended as a guide for grduate students preparing for qualifying exams.

F. E. J. Linton Source: CHOICE