实分析和抽象分析

内容简介

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

目录

ChapterOne:SetTheoryandAlgebra Section1.Thealgebraofsets Section2.Relationsandfunctions Section3.Theaxiomofchoiceandsomeequivalents Section4.Cardinalnumbersandordinalnumbers Section5.Constructionoftherealandcomplexnumberfields ChapterTwo:TopologyandContinuousFunctions Section6.Topologicalpreliminaries Section7.Spacesofcontinuousfunctions ChapterThree:TheLebesgueIntegral Section8.TheRiemann-Stieltjesintegral Section9.Extendingcertainfunctionals Section10.Measuresandmeasurablesets Section11.Measurablefunctions Section12.TheabstractLebesgueintegral ChapterFour:FunctionSpacesandBanachSpaces Section13.Thespaces Section14.AbstractBanachspaces Section15.Theconjugatespaceof(1Section16.AbstractHilbertspaces ChapterFive:Differentiation Section17.Differentiableandnondifferentiablefunctions Section18.Absolutelycontinuousfunctions Section19.ComplexmeasuresandtheLEBESGUE-RADON-NIKODYMtheorem Section20.ApplicationsoftheLEBESGUE-RADON-NIKODYMtheorem ChapterSix:IntegrationonProductSpaces Section21.Theproductoftwomeasurespaces Section22.Productsofinfinitelymanymeasurespaces IndexofSymbols IndexofAuthorsandTerms