# Basic Real Analysis. Along with a companion volume Advanced real analysis

Both these books appear in the series called ‘Cornerstones’. In the preface of the first book the author says, “This book and its companion Advanced Real Analysis systematically develop concepts and tools in real analysis that are vital to every mathematician, whether pure or applied, aspiring or established. The two books together contain what the young mathematician needs to know about real analysis in order to communicate well with colleagues in all branches of mathematics”. The textbook has an encyclopedic character; it is carefully written and contains almost all one can imagine after reading the statement quoted above. There are some exceptions, e.g. complex analysis and surface integration are not included. After a short review of the basics of calculus of one real variable, the first book deals with metric spaces (ca 650 pp.), calculus of several variables and the theory of ordinary differential equations and their systems. This part is followed by an exposition of measure theory, Lebesgue integration and its differentiation on the real line and the Fourier transform in Euclidean space. After a treatment of Lp spaces, topological spaces and a chapter on integration in locally compact spaces, the first book ends with a chapter devoted to Banach and Hilbert spaces.

The second book starts with an introduction to boundary-value problems. Just a list of chapters does not give a complete picture; for example in a chapter on Euclidean Fourier analysis, one can find tempered distributions, Sobolev spaces, harmonic functions and Hp theory, the Calderon-Zygmund theorem with its applications, and multiple Fourier series. From the remaining chapters, we point out some topics: distributions, Fourier analysis and compact groups, and pseudodifferential operators on manifolds. The book (at about 460 pages) closes with a chapter on the foundations of probability, which includes the strong law of large numbers. Each chapter is followed by a number of problems (altogether 306 + 190), which often include further important results. All problems are provided with hints at the end of both books and the first book also contains ten appendices on facts that a reader might like to recall without searching through the literature. Well-balanced indexes, bibliographical notes, abstracts at the beginning of each chapter, schema of dependence among chapters and a guide for the reader make both books easier to handle, but they require not only reading but also active cooperation of the reader. Both books can be recommended for the mathematical libraries of universities.

**Submitted by Anonymous |

**21 / Oct / 2011