A Guide to Advanced Real Analysis

This concise guide to real analysis covers the core material of a graduate level real analysis course. 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. The prerequisite is a familiarity with classical real-variable theory.

• Covers the core material found on a graduate course on real analysis
• Gives an overview of the subject so that it can be used as a guide for a beginner or as a refresher for those who have previously studied the subject
• To remain concise, essential definitions, major theorems, and key ideas of proofs are included and technical details are not