實分析和抽象分析

內(nèi)容概要

　　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 customarily 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 ave given elementary avatars of all important theorems， with appropriate suggestions for skipping. We have given complete definitions， explanations， and proofs throughout， so that the book should be usable for individual study as well as for a course text

書籍目錄

Chapter One: Set Theory and Algebra  Section 1.  The algebra of sets  Section 2.  Relations and functions  Section 3.  The axiom of choice and ordinal numbers  Section 4.  Cardinal numbers and ordinal numbers  Section 5.  Construction of the real and complex number fieldsChapter Two: Topology and Continuous Functions  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 integralChapter Four: Function Spaces and Banach Spaces  Section 13.  The spaces  Section 14.  Abstract Banach spaces  Section 15.  The conjugate space  Section 16.  Abstract Hilbert spacesChapter Five:  Differentiation  Section 17.  Differentiable and nondifferentiable functions  Section 18.  Absolutely continuous functions  Section 19.  Complex measures and the LEBESGUE-RADON-NIKODYM theorem  Section 20.  Applications of the LEBESGUE-RADON-NIKODYM theoremChapter Six:  Integration on Product Spaces  Section 21.  The product of two measure spaces  Section 22.  Products of infinitely many measure spacesIndex of SymbolsIndex of Authors and Terms

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