Real and Abstract Analysis, Softcover reprint of the original 1st ed. 1965 A modern treatment of the theory of functions of a real variable

This book is first of all designed as a text for the course usually called "theory of functions of a real variable". This course is at present cus­ tomarily offered as a first or second year graduate course in United States universities, although there are signs that this sort of analysis will soon penetrate upper division undergraduate curricula. We have included every topic that we think essential for the training of analysts, and we have also gone down a number of interesting bypaths. We hope too that the book will be useful as a reference for mature mathematicians and other scientific workers. Hence we have presented very general and complete versions of a number of important theorems and constructions. Since these sophisticated versions may be difficult for the beginner, we have given elementary avatars of all important theorems, with appro­ priate suggestions for skipping. We have given complete definitions, ex­ planations, and proofs throughout, so that the book should be usable for individual study as well as for a course text. Prerequisites for reading the book are the following. The reader is assumed to know elementary analysis as the subject is set forth, for example, in TOM M. ApOSTOL'S Mathematical Analysis [Addison-Wesley Publ. Co., Reading, Mass., 1957], or WALTER RUDIN'S Principles of M athe­ nd matical Analysis [2 Ed., McGraw-Hill Book Co., New York, 1964].

One: Set Theory and Algebra.- Section 1. The algebra of sets.- Section 2. Relations and functions.- Section 3. The axiom of choice and some equivalents.- Section 4. Cardinal numbers and ordinal numbers.- Section 5. Construction of the real and complex number fields.- Two: Topology and Continuous Functions.- Section 6. Topological preliminaries.- Section 7. Spaces of continuous functions.- Three: The Lebesgue Integral.- Section 8. The Riemann-Stieltjes integral.- Section 9. Extending certain functionals.- Section 10. Measures and measurable sets.- Section 11. Measurable functions.- Section 12. The abstract Lebesgue integral.- Four: Function Spaces and Banach Spaces.- Section 13. The spaces $${\mathcal{L}_p}(1 \leqq p < \infty )$$.- Section 14. Abstract Banach spaces.- Section 15. The conjugate space of $${\mathcal{L}_p}(1 < p < \infty )$$.- Section 16. Abstract Hilbert spaces.- Five: Differentiation.- Section 17. Differentiable and nondifferentiable functions.- Section 18. Absolutely continuous functions.- Section 19. Complex measures and the Lebesgue-Radon-Nikodým theorem.- Section 20. Applications of the Lebesgue-Radon-Nikodým theorem.- Six: Integration on Product Spaces.- Section 21. The product of two measure spaces.- Section 22. Products of infinitely many measure spaces.- Index of Symbols.- Index of Authors and Terms.