Real analysis

the long version

"Real Analysis "is written for the senior under-graduate or first-year" "graduate student who has a workable knowledge of advanced cal-culus. Chapter 1 contains that part of the set theory which underlies all the material in this book. Chapter 2 is at once a review and an introduction to the succeeding chapters. Chapter 3 provides a detailed presentation of the Lebesgue measure and integral. Chapter 4 demonstrates the relation between differentiation and integration in relation to Chapter 3. Chapters 5 and 7 contain the basic mater-ial which derives from the theory on metric spaces and topological spaces. The topics discussed in these chapters are most perti-nent to analysis. Chapter 6 deals with the classical Banach spaces and can be viewed as an introduction to Chapter 8, which con-siders abstract Banach spaces. Chapter 9 is entirely devoted to Hilbert space, which occupies a special position among other Banach spaces. Because of its importance, the chapter presents introduc-tory material and some fundamental results in this area. Chapter 10 focuses upon the abstraction of the most important properties of the Lebesgue measure and Lebesgue inte-gral. Each chapter is followed by a Problem section. The book contains a bibliography.