Principles of Analysis

The author's aim for the book under review is to provide a rigorous and detailed treatment of the essentials of measure and integration, as well as other topics in functional analysis at the graduate level. Although he assumes readers to have an undergraduate background, such as real analysis (including some experience in dealing with limits, continuity, di erentiation, Riemann integration, and uniform convergence, including elementary set theory), a standard course of complex analysis (function theory, Cauchy's integral equation), and a knowledge of basic linear algebra, this book could also be very useful for a reader with a weaker mathematical background. This is possible since the excellently constructed introduction in Chapter 0 is a very good base for systematizing and developing the mathematical background for a broad group of readers. The book is divided into four parts. In Part I, which consists of Chapters 1{7, the author develops a detailed course concerning the general theory of Lebesgue integration as well as Fourier analysis on Rd (Chapter 6) and measures on locally compact spaces (Chapter 7). A short course on the general theory of Lebesgue integration could be based on Chapters 1{5 only but the full variant looks more attractive. It must be noted that the author's exposition is on a very high level as well as very clear and easily understandable. Part II is presented as a course in functional analysis. The author considers Chapters 8{12 to be the core of such a course. Chapter 13 could be an optional choice, but can be also included in the course. Chapter 14 plays an important role concerning Part I and Part II. This chapter includes not only deeper theorems in functional analysis but also several well-chosen applications. Note that some of them are related to the measure and integration developed in Part I and the others with the applications in the remainder of the book. Part III (Chapters 15{17) is a key part in the book since it includes many topics and applications that depend on, and indeed are meant to illustrate, the power of topics developed in the first two parts. It must be noted that these chapters are almost independent. Their goal is to provide a relatively quick overview of the subjects treated therein. The detailed exposition that this approach allows means that the reader can follow the development with relative ease. In addition to allowing the reader to consult the themes considered, some specialized sources are listed in the bibliography. Part IV consists of two appendices with proofs of the change of variables theorem and a theorem on separate and joint continuity. A reader may choose to safely omit the proofs without disturbing the flow of the text, as the author notes. An advantage for the readers is that the book contains a lot of exercises (nearly 700). It is very convenient that hints and/or a framework of intermediate steps are given for the more di cult exercises. Many of these are extensions of material in the text or are of special independent interest. Additionally, the exercises related in a critical way to material elsewhere in the text are marked with either an upward arrow, referring to earlier results, or a downward arrow, referring to later material. Instructors may obtain complete solutions to the exercises from the publisher. In conclusion, I strongly recommend the book because it will be helpful for every level of reader. I only regret that it was not written when I was a student. - Andrey I. Zahariev - Mathematical Reviews Clippings February 2019