BBT Real Analysis Bruckner, Bruckner, Thomson Second Edition
Table of Contents
>>{Preface}{xiii}{chapter*.1}
>>..{1}{Background and Preview}}{1}{chapter.1}
>>..{1.1}The Real Numbers }{2}{section.1.1}
>>..{1.2}Compact Sets of Real Numbers }{10}{section.1.2}
>>..{1.3}Countable Sets }{13}{section.1.3}
>>..{1.4}Uncountable Cardinals }{18}{section.1.4}
>>..{1.5}Transfinite Ordinals }{21}{section.1.5}
>>..{1.6}Category }{25}{section.1.6}
>>..{1.7}Outer Measure and Outer Content }{29}{section.1.7}
>>..{1.8}Small Sets }{32}{section.1.8}
>>..{1.9}Measurable Sets of Real Numbers }{37}{section.1.9}
>>..{1.10}Nonmeasurable Sets }{42}{section.1.10}
>>..{1.11}Zorn's Lemma }{47}{section.1.11}
>>..{1.12}Borel Sets of Real Numbers }{49}{section.1.12}
>>..{1.13}Analytic Sets of Real Numbers}{51}{section.1.13}
>>..{1.14}Bounded Variation }{53}{section.1.14}
>>..{1.15}Newton's Integral }{58}{section.1.15}
>>..{1.16}Cauchy's Integral }{59}{section.1.16}
>>..{1.17}Riemann's Integral }{62}{section.1.17}
>>..{1.18}Volterra's Example }{65}{section.1.18}
>>..{1.19}Riemann--Stieltjes Integral }{68}{section.1.19}
>>..{1.20}Lebesgue's Integral }{73}{section.1.20}
>>..{1.21}The Generalized Riemann Integral }{76}{section.1.21}
>>..{1.22}Additional Problems for Chapter\nobreakspace {}1}{79}{section.1.22}
>>..{2}{Measure Spaces}}{85}{chapter.2}
>>..{2.1}One-Dimensional Lebesgue Measure }{86}{section.2.1}
>>..{2.2}Additive Set Functions}{93}{section.2.2}
>>..{2.3}Measures and Signed Measures }{101}{section.2.3}
>>..{2.4}Limit Theorems}{106}{section.2.4}
>>..{2.5}Jordan and Hahn Decomposition }{110}{section.2.5}
>>..{2.6}Complete Measures }{115}{section.2.6}
>>..{2.7}Outer Measures }{118}{section.2.7}
>>..{2.8}Method\nobreakspace {}I}{124}{section.2.8}
>>..{2.9}Regular Outer Measures }{128}{section.2.9}
>>..{2.10}Nonmeasurable Sets}{133}{section.2.10}
>>..{2.11}More About Method\nobreakspace {}I }{137}{section.2.11}
>>..{2.12}Completions}{141}{section.2.12}
>>..{2.13}Additional Problems for Chapter\nobreakspace {}2}{145}{section.2.13}
>>..{3}{Metric Outer Measures}}{149}{chapter.3}
>>..{3.1}Metric Space}{150}{section.3.1}
>>..{3.2}Metric Outer Measures }{155}{section.3.2}
>>..{3.3}Method II}{161}{section.3.3}
>>..{3.4}Approximations}{166}{section.3.4}
>>..{3.5}Construction of Lebesgue--Stieltjes Measures}{170}{section.3.5}
>>..{3.6}Properties of Lebesgue--Stieltjes Measures}{178}{section.3.6}
>>..{3.7}Lebesgue--Stieltjes Measures in $\unhbox \voidb@x \hbox {I\kern -.1667em\unhbox \voidb@x \hbox {R}}{}^n$}{184}{section.3.7}
>>..{3.8}Hausdorff Measures and Hausdorff Dimension}{187}{section.3.8}
>>..{3.9}Methods III and IV}{198}{section.3.9}
>>..{3.10}Additional Remarks}{203}{section.3.10}
>>..{3.11}Additional Problems for Chapter\nobreakspace {}3}{209}{section.3.11}
>>..{4}{Measurable Functions}}{215}{chapter.4}
>>..{4.1}Definitions and Basic Properties }{216}{section.4.1}
>>..{4.2}Sequences of Measurable Functions }{224}{section.4.2}
>>..{4.3}Egoroff's Theorem }{230}{section.4.3}
>>..{4.4}Approximations by Simple Functions}{234}{section.4.4}
>>..{4.5}Approximation by Continuous Functions}{240}{section.4.5}
>>..{4.6}Additional Problems for Chapter\nobreakspace {}4}{246}{section.4.6}
>>..{5}{Integration}}{251}{chapter.5}
>>..{5.1}Introduction}{252}{section.5.1}
>>..{5.2}Integrals of Nonnegative Functions}{257}{section.5.2}
>>..{5.3}Fatou's Lemma}{264}{section.5.3}
>>..{5.4}Integrable Functions}{269}{section.5.4}
>>..{5.5}Riemann and Lebesgue}{274}{section.5.5}
>>..{5.6}Countable Additivity of the Integral}{286}{section.5.6}
>>..{5.7}Absolute Continuity}{289}{section.5.7}
>>..{5.8}Radon--Nikodym Theorem}{296}{section.5.8}
>>..{5.9}Convergence Theorems}{306}{section.5.9}
>>..{5.10}Relations to Other Integrals}{314}{section.5.10}
>>..{5.11}Integration of Complex Functions}{321}{section.5.11}
>>..{5.12}Additional Problems for Chapter\nobreakspace {}5}{326}{section.5.12}
>>..{6}{Fubini's Theorem}}{332}{chapter.6}
>>..{6.1}Product Measures}{334}{section.6.1}
>>..{6.2}Fubini's Theorem}{344}{section.6.2}
>>..{6.3}Tonelli's Theorem}{348}{section.6.3}
>>..{6.4}Additional Problems for Chapter\nobreakspace {}6}{350}{section.6.4}
>>..{7}{Differentiation}}{353}{chapter.7}
>>..{7.1}The Vitali Covering Theorem}{354}{section.7.1}
>>..{7.2}Functions of Bounded Variation}{361}{section.7.2}
>>..{7.3}The Banach--Zarecki Theorem}{366}{section.7.3}
>>..{7.4}Determining a Function by Its Derivative}{371}{section.7.4}
>>..{7.5}Calculating a Function from Its Derivative}{374}{section.7.5}
>>..{7.6}Total Variation of a Continuous Function}{384}{section.7.6}
>>..{7.7}VBG$_*$ Functions}{391}{section.7.7}
>>..{7.8}Approximate Continuity, Lebesgue\ Points}{397}{section.7.8}
>>..{7.9}Additional Problems for Chapter\nobreakspace {}7}{405}{section.7.9}
>>..{8}{Differentiation of Measures}}{414}{chapter.8}
>>..{8.1}Differentiation of Lebesgue--Stieltjes\ Measures}{415}{section.8.1}
>>..{8.2}The Cube Basis; Ordinary Differentiation}{421}{section.8.2}
>>..{8.3}The Lebesgue Decomposition Theorem}{428}{section.8.3}
>>..{8.4}The Interval Basis; Strong Differentiation}{431}{section.8.4}
>>..{8.5}Net Structures}{440}{section.8.5}
>>..{8.6}Radon--Nikodym\ Derivative in a Measure Space}{448}{section.8.6}
>>..{8.7}Summary, Comments, and References}{459}{section.8.7}
>>..{8.8}Additional Problems for Chapter\nobreakspace {}8}{463}{section.8.8}
>>..{9}{Metric Spaces}}{466}{chapter.9}
>>..{9.1}Definitions and Examples}{467}{section.9.1}
>>..{9.2}Convergence and Related Notions}{478}{section.9.2}
>>..{9.3}Continuity}{482}{section.9.3}
>>..{9.4}Homeomorphisms and Isometries}{488}{section.9.4}
>>..{9.5}Separable Spaces}{493}{section.9.5}
>>..{9.6}Complete Spaces}{495}{section.9.6}
>>..{9.7}Contraction Maps}{503}{section.9.7}
>>..{9.8}Applications of Contraction Mappings}{505}{section.9.8}
>>..{9.9}Compactness}{514}{section.9.9}
>>..{9.10}Totally Bounded Spaces}{518}{section.9.10}
>>..{9.11}Compact Sets in $\unhbox \voidb@x \hbox {$\cal C$}(X)$}{521}{section.9.11}
>>..{9.12}Application of the Arzel\`a--Ascoli Theorem}{525}{section.9.12}
>>..{9.13}The Stone--Weierstrass Theorem}{529}{section.9.13}
>>..{9.14}The Isoperimetric Problem}{532}{section.9.14}
>>..{9.15}More on Convergence}{537}{section.9.15}
>>..{9.16}Additional Problems for Chapter\nobreakspace {}9}{541}{section.9.16}
>>..{10}{Baire Category}}{546}{chapter.10}
>>..{10.1}The Baire Category Theorem}{547}{section.10.1}
>>..{10.2}The Banach--Mazur Game}{554}{section.10.2}
>>..{10.3}The First Classes of Baire and Borel}{560}{section.10.3}
>>..{10.4}Properties of Baire-1 Functions}{568}{section.10.4}
>>..{10.5}Topologically Complete Spaces}{573}{section.10.5}
>>..{10.6}Applications to Function Spaces}{579}{section.10.6}
>>..{10.7}Additional Problems for Chapter\nobreakspace {}10}{594}{section.10.7}
>>..{11}{Analytic Sets}}{600}{chapter.11}
>>..{11.1}Products of Metric Spaces }{601}{section.11.1}
>>..{11.2}Baire Space }{603}{section.11.2}
>>..{11.3}Analytic Sets }{607}{section.11.3}
>>..{11.4}Borel Sets }{612}{section.11.4}
>>..{11.5}An Analytic Set That Is Not Borel }{617}{section.11.5}
>>..{11.6}Measurability of Analytic Sets }{620}{section.11.6}
>>..{11.7}The Suslin Operation }{623}{section.11.7}
>>..{11.8}A Method to Show a Set Is Not Borel}{626}{section.11.8}
>>..{11.9}Differentiable Functions}{630}{section.11.9}
>>..{11.10}Additional Problems for Chapter\nobreakspace {}11}{636}{section.11.10}
>>..{12}{Banach Spaces}}{638}{chapter.12}
>>..{12.1}Normed Linear Spaces}{639}{section.12.1}
>>..{12.2}Compactness}{646}{section.12.2}
>>..{12.3}Linear Operators}{651}{section.12.3}
>>..{12.4}Banach Algebras}{656}{section.12.4}
>>..{12.5}The Hahn--Banach Theorem}{661}{section.12.5}
>>..{12.6}Improving Lebesgue\ Measure}{667}{section.12.6}
>>..{12.7}The Dual Space}{675}{section.12.7}
>>..{12.8}The Riesz Representation Theorem}{679}{section.12.8}
>>..{12.9}Separation of Convex Sets}{686}{section.12.9}
>>..{12.10}An Embedding Theorem}{692}{section.12.10}
>>..{12.11}The Uniform Boundedness Principle}{696}{section.12.11}
>>..{12.12}An Application to Summability}{700}{section.12.12}
>>..{12.13}The Open Mapping Theorem}{706}{section.12.13}
>>..{12.14}The Closed Graph Theorem}{711}{section.12.14}
>>..{12.15}Additional Problems for Chapter\nobreakspace {}12}{714}{section.12.15}
>>..{13}{The $L_p$ spaces }}{717}{chapter.13}
>>..{13.1}The Basic Inequalities}{718}{section.13.1}
>>..{13.2}The $\ell _p$ and $L_p$ Spaces $(1\leq p< \infty )$}{723}{section.13.2}
>>..{13.3}The Spaces $\ell _\infty $ and $L_\infty $ }{726}{section.13.3}
>>..{13.4}Separability}{729}{section.13.4}
>>..{13.5}The Spaces $\ell _2$ and $L_2$}{732}{section.13.5}
>>..{13.6}Continuous Linear Functionals}{740}{section.13.6}
>>..{13.7}The $L_p$ Spaces $(0>>..{13.8}Relations}{748}{section.13.8}
>>..{13.9}The Banach Algebra $L_1(\unhbox \voidb@x \hbox {I\kern -.1667em\unhbox \voidb@x \hbox {R}})$}{752}{section.13.9}
>>..{13.10}Weak Sequential Convergence}{760}{section.13.10}
>>..{13.11}Closed Subspaces of the $L_p$ Spaces}{764}{section.13.11}
>>..{13.12}Additional Problems for Chapter\nobreakspace {}13}{768}{section.13.12}
>>..{14}{Hilbert Spaces}}{771}{chapter.14}
>>..{14.1}Inner Products }{772}{section.14.1}
>>..{14.2}Convex Sets}{780}{section.14.2}
>>..{14.3}Continuous Linear Functionals}{783}{section.14.3}
>>..{14.4}Orthogonal Series}{786}{section.14.4}
>>..{14.5}Weak Sequential Convergence}{794}{section.14.5}
>>..{14.6}Compact Operators}{800}{section.14.6}
>>..{14.7}Projections}{804}{section.14.7}
>>..{14.8}Eigenvectors and Eigenvalues}{808}{section.14.8}
>>..{14.9}Spectral Decomposition}{814}{section.14.9}
>>..{14.10}Additional Problems for Chapter\nobreakspace {}14}{819}{section.14.10}
>>..{15}{Fourier Series}}{823}{chapter.15}
>>..{15.1}Notation and Terminology}{825}{section.15.1}
>>..{15.2}Dirichlet's Kernel}{831}{section.15.2}
>>..{15.3}Fej\'{e}r's Kernel}{836}{section.15.3}
>>..{15.4}Convergence of the Ces\`{a}ro Means}{840}{section.15.4}
>>..{15.5}The Fourier Coefficients}{846}{section.15.5}
>>..{15.6}Weierstrass Approximation Theorem}{849}{section.15.6}
>>..{15.7}Pointwise Convergence: Jordan's Test }{853}{section.15.7}
>>..{15.8}Pointwise Convergence: Dini's Test }{860}{section.15.8}
>>..{15.9}Pointwise Divergence}{863}{section.15.9}
>>..{15.10}Characterizations}{866}{section.15.10}
>>..{15.11}Fourier Series in Hilbert Space}{869}{section.15.11}
>>..{15.12}Riemann's Theorems}{873}{section.15.12}
>>..{15.13}Cantor's Uniqueness Theorem}{878}{section.15.13}
>>..{15.14}Additional Problems for Chapter\nobreakspace {}15}{883}{section.15.14}
>>{Index}{885}{Item.1868}