logo资料库

Real analysis.pdf

发布时间：2022-06-08 发布人：admin 分类：说明书 资料大小：4.96M 资料格式：pdf 举报 版权申诉
第1页 / 共921页
第2页 / 共921页
第3页 / 共921页
第4页 / 共921页
第5页 / 共921页
第6页 / 共921页
第7页 / 共921页
第8页 / 共921页
资料共921页，剩余部分请下载后查看
    Preface
    Background and Preview
    The Real Numbers
    Compact Sets of Real Numbers
    Countable Sets
    Uncountable Cardinals
    Transfinite Ordinals
    Category
    Outer Measure and Outer Content
    Small Sets
    Measurable Sets of Real Numbers
    Nonmeasurable Sets
    Zorn's Lemma
    Borel Sets of Real Numbers
    Analytic Sets of Real Numbers
    Bounded Variation
    Newton's Integral
    Cauchy's Integral
    Riemann's Integral
    Volterra's Example
    Riemann--Stieltjes Integral
    Lebesgue's Integral
    The Generalized Riemann Integral
    Additional Problems for Chapter 1
    Measure Spaces
    One-Dimensional Lebesgue Measure
    Additive Set Functions
    Measures and Signed Measures
    Limit Theorems
    Jordan and Hahn Decomposition
    Complete Measures
    Outer Measures
    Method I
    Regular Outer Measures
    Nonmeasurable Sets
    More About Method I
    Completions
    Additional Problems for Chapter 2
    Metric Outer Measures
    Metric Space
    Metric Outer Measures
    Method II
    Approximations
    Construction of Lebesgue--Stieltjes Measures
    Properties of Lebesgue--Stieltjes Measures
    Lebesgue--Stieltjes Measures in IRn
    Hausdorff Measures and Hausdorff Dimension
    Methods III and IV
    Additional Remarks
    Additional Problems for Chapter 3
    Measurable Functions
    Definitions and Basic Properties
    Sequences of Measurable Functions
    Egoroff's Theorem
    Approximations by Simple Functions
    Approximation by Continuous Functions
    Additional Problems for Chapter 4
    Integration
    Introduction
    Integrals of Nonnegative Functions
    Fatou's Lemma
    Integrable Functions
    Riemann and Lebesgue
    Countable Additivity of the Integral
    Absolute Continuity
    Radon--Nikodym Theorem
    Convergence Theorems
    Relations to Other Integrals
    Integration of Complex Functions
    Additional Problems for Chapter 5
    Fubini's Theorem
    Product Measures
    Fubini's Theorem
    Tonelli's Theorem
    Additional Problems for Chapter 6
    Differentiation
    The Vitali Covering Theorem
    Functions of Bounded Variation
    The Banach--Zarecki Theorem
    Determining a Function by Its Derivative
    Calculating a Function from Its Derivative
    Total Variation of a Continuous Function
    VBG* Functions
    Approximate Continuity, Lebesgue Points
    Additional Problems for Chapter 7
    Differentiation of Measures
    Differentiation of Lebesgue--Stieltjes Measures
    The Cube Basis; Ordinary Differentiation
    The Lebesgue Decomposition Theorem
    The Interval Basis; Strong Differentiation
    Net Structures
    Radon--Nikodym Derivative in a Measure Space
    Summary, Comments, and References
    Additional Problems for Chapter 8
    Metric Spaces
    Definitions and Examples
    Convergence and Related Notions
    Continuity
    Homeomorphisms and Isometries
    Separable Spaces
    Complete Spaces
    Contraction Maps
    Applications of Contraction Mappings
    Compactness
    Totally Bounded Spaces
    Compact Sets in C(X)
    Application of the Arzelà--Ascoli Theorem
    The Stone--Weierstrass Theorem
    The Isoperimetric Problem
    More on Convergence
    Additional Problems for Chapter 9
    Baire Category
    The Baire Category Theorem
    The Banach--Mazur Game
    The First Classes of Baire and Borel
    Properties of Baire-1 Functions
    Topologically Complete Spaces
    Applications to Function Spaces
    Additional Problems for Chapter 10
    Analytic Sets
    Products of Metric Spaces
    Baire Space
    Analytic Sets
    Borel Sets
    An Analytic Set That Is Not Borel
    Measurability of Analytic Sets
    The Suslin Operation
    A Method to Show a Set Is Not Borel
    Differentiable Functions
    Additional Problems for Chapter 11
    Banach Spaces
    Normed Linear Spaces
    Compactness
    Linear Operators
    Banach Algebras
    The Hahn--Banach Theorem
    Improving Lebesgue Measure
    The Dual Space
    The Riesz Representation Theorem
    Separation of Convex Sets
    An Embedding Theorem
    The Uniform Boundedness Principle
    An Application to Summability
    The Open Mapping Theorem
    The Closed Graph Theorem
    Additional Problems for Chapter 12
    The Lp spaces
    The Basic Inequalities
    The p and Lp Spaces (1p< )
    The Spaces and L
    Separability
    The Spaces 2 and L2
    Continuous Linear Functionals
    The Lp Spaces (0
    Relations
    The Banach Algebra L1(IR)
    Weak Sequential Convergence
    Closed Subspaces of the Lp Spaces
    Additional Problems for Chapter 13
    Hilbert Spaces
    Inner Products
    Convex Sets
    Continuous Linear Functionals
    Orthogonal Series
    Weak Sequential Convergence
    Compact Operators
    Projections
    Eigenvectors and Eigenvalues
    Spectral Decomposition
    Additional Problems for Chapter 14
    Fourier Series
    Notation and Terminology
    Dirichlet's Kernel
    Fejér's Kernel
    Convergence of the Cesàro Means
    The Fourier Coefficients
    Weierstrass Approximation Theorem
    Pointwise Convergence: Jordan's Test
    Pointwise Convergence: Dini's Test
    Pointwise Divergence
    Characterizations
    Fourier Series in Hilbert Space
    Riemann's Theorems
    Cantor's Uniqueness Theorem
    Additional Problems for Chapter 15
    Index
    REAL ANALYSIS ClassicalRealAnalysis.com
    REAL ANALYSIS ————————————— Bruckner2·Thomson ————————————— Andrew M. Bruckner Judith B. Bruckner Brian S. Thomson
    ii www.classicalrealanalysis.com This PDF file is for the text Real Analysis originally published by Prentice Hall (Pearson) in 1997. The paging is slightly different from the original printed version and may be different from earlier PDF files distributed. The authors retain the copyright and all commercial uses. [2007] Date PDF file compiled: January 30, 2008. REAL ANALYSIS ClassicalRealAnalysis.com
    Library of Congress Cataloging-in-Publication Data Bruckner, Andrew M. Real Analysis. / Andrew M. Bruckner, Judith B. Bruckner, Brian S. Thomson. p. cm. Includes index. ISBN: 0-13-458886-X (hardcover : alk. paper) 1. Mathematical analysis. 2. Functions of real variables. I. Bruckner, Judith B. II. Thomson, Brian S. III. Title. QA300.B74 1997 515 .8–dc20 96–22123 CIP Acquisitions editor: George Lobell Editorial Director: Tim Bozik Editorial Director: Jerome Grant AVP, Production and Manufacturing: David W. Riccardi Production Editor: Elaine Wetterau Managing Editor: Linda Mihatov Behrens Marketing Manager: John Tweedale Creative Director: Paula Maylahn Art Director: Jayne Conte Cover Designer: Bruce Kenselaar Manufacturing Buyer: Alan Fischer Manufacturing Manager: Trudy Pisciotti Editorial Assistant: Gale Epps Cover Photograph: Carmine M. Saccardo cOriginal copyright 1997 Prentice-Hall, Inc. The authors now hold the copyright and retain all rights. All rights reserved. No part of this book may be reproduced, in any form or by any means, without permission in writing from the authors. REAL ANALYSIS ClassicalRealAnalysis.com
    iv Originally printed in the United States of America 10 9 6 5 8 7 4 3 2 1 ISBN: 0-13-458886-X Prentice-Hall International (UK) Limited, London Prentice-Hall International (UK) Limited, London Prentice-Hall of Australia Pty. Limited, Sydney Prentice-Hall Canada, Inc., (UK) Limited, Toronto Prentice-Hall Hispanoamericana, S.A., Mexico Prentice-Hall of India Private Limited, New Delhi Prentice-Hall of Japan, Inc., Tokyo Simon & Schuster Asia Pte. Ltd., Singapore Editora Prentice-Hall do Brasil, Ltda., London REAL ANALYSIS ClassicalRealAnalysis.com
    Contents Preface 1 Background and Preview 1.1 The Real Numbers 1.2 Compact Sets of Real Numbers 1.3 Countable Sets 1.4 Uncountable Cardinals 1.5 Transfinite Ordinals 1.6 Category 1.7 Outer Measure and Outer Content 1.8 1.9 Measurable Sets of Real Numbers 1.10 Nonmeasurable Sets 1.11 Zorn’s Lemma 1.12 Borel Sets of Real Numbers 1.13 Analytic Sets of Real Numbers 1.14 Bounded Variation 1.15 Newton’s Integral Small Sets xiii 1 2 10 13 18 21 25 29 32 37 42 47 49 51 53 58 v REAL ANALYSIS ClassicalRealAnalysis.com
    vi Contents 1.16 Cauchy’s Integral 1.17 Riemann’s Integral 1.18 Volterra’s Example 1.19 Riemann–Stieltjes Integral 1.20 Lebesgue’s Integral 1.21 The Generalized Riemann Integral 1.22 Additional Problems for Chapter 1 2 Measure Spaces Jordan and Hahn Decomposition 2.1 One-Dimensional Lebesgue Measure 2.2 Additive Set Functions 2.3 Measures and Signed Measures 2.4 Limit Theorems 2.5 2.6 Complete Measures 2.7 Outer Measures 2.8 Method I 2.9 Regular Outer Measures 2.10 Nonmeasurable Sets 2.11 More About Method I 2.12 Completions 2.13 Additional Problems for Chapter 2 3 Metric Outer Measures 3.1 Metric Space 3.2 Metric Outer Measures 3.3 Method II 3.4 Approximations 3.5 Construction of Lebesgue–Stieltjes Measures 59 62 65 68 73 76 79 85 86 93 101 106 110 115 118 124 128 133 137 141 145 149 150 155 161 166 170 REAL ANALYSIS ClassicalRealAnalysis.com
    Contents 3.6 Properties of Lebesgue–Stieltjes Measures 3.7 Lebesgue–Stieltjes Measures in IRn 3.8 Hausdorff Measures and Hausdorff Dimension 3.9 Methods III and IV 3.10 Additional Remarks 3.11 Additional Problems for Chapter 3 4 Measurable Functions Sequences of Measurable Functions 4.1 Definitions and Basic Properties 4.2 4.3 Egoroff’s Theorem 4.4 Approximations by Simple Functions 4.5 Approximation by Continuous Functions 4.6 Additional Problems for Chapter 4 5 Integration Integrable Functions Introduction Integrals of Nonnegative Functions 5.1 5.2 5.3 Fatou’s Lemma 5.4 5.5 Riemann and Lebesgue 5.6 Countable Additivity of the Integral 5.7 Absolute Continuity 5.8 Radon–Nikodym Theorem 5.9 Convergence Theorems 5.10 Relations to Other Integrals 5.11 Integration of Complex Functions 5.12 Additional Problems for Chapter 5 vii 178 184 187 198 203 209 215 216 224 230 234 240 246 251 252 257 264 269 274 286 289 296 306 314 321 326 REAL ANALYSIS ClassicalRealAnalysis.com
    分享到：
    赞 收藏

    相关推荐

    资料库

    课程资源

    共收录17145份资料，累计13个分类，关注成员有19位，主要包括：PHP,网络管理,网页制作,Java,.Net,数据库,3G/移动开发,C/C++,游戏开发,嵌入式,讲义,软件测试,专业指导

    热门标签

    PHP 网络管理 网页制作 Java .Net 数据库 3G/移动开发 C/C++ 游戏开发 嵌入式 讲义 软件测试 专业指导

    最新资料