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Real and Complex Analysis


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Table of Contents

PrefacePrologue: The Exponential FunctionChapter 1: Abstract IntegrationSet-theoretic notations and terminologyThe concept of measurabilitySimple functionsElementary properties of measuresArithmetic in [0, ]Integration of positive functionsIntegration of complex functionsThe role played by sets of measure zeroExercisesChapter 2: Positive Borel MeasuresVector spacesTopological preliminariesThe Riesz representation theoremRegularity properties of Borel measuresLebesgue measureContinuity properties of measurable functionsExercisesChapter 3: Lp-SpacesConvex functions and inequalitiesThe Lp-spacesApproximation by continuous functionsExercisesChapter 4: Elementary Hilbert Space TheoryInner products and linear functionalsOrthonormal setsTrigonometric seriesExercisesChapter 5: Examples of Banach Space TechniquesBanach spacesConsequences of Baire's theoremFourier series of continuous functionsFourier coefficients of L1-functionsThe Hahn-Banach theoremAn abstract approach to the Poisson integralExercisesChapter 6: Complex MeasuresTotal variationAbsolute continuityConsequences of the Radon-Nikodym theoremBounded linear functionals on LpThe Riesz representation theoremExercisesChapter 7: DifferentiationDerivatives of measuresThe fundamental theorem of CalculusDifferentiable transformationsExercisesChapter 8: Integration on Product SpacesMeasurability on cartesian productsProduct measuresThe Fubini theoremCompletion of product measuresConvolutionsDistribution functionsExercisesChapter 9: Fourier TransformsFormal propertiesThe inversion theoremThe Plancherel theoremThe Banach algebra L1ExercisesChapter 10: Elementary Properties of Holomorphic FunctionsComplex differentiationIntegration over pathsThe local Cauchy theoremThe power series representationThe open mapping theoremThe global Cauchy theoremThe calculus of residuesExercisesChapter 11: Harmonic FunctionsThe Cauchy-Riemann equationsThe Poisson integralThe mean value propertyBoundary behavior of Poisson integralsRepresentation theoremsExercisesChapter 12: The Maximum Modulus PrincipleIntroductionThe Schwarz lemmaThe Phragmen-Lindeloef methodAn interpolation theoremA converse of the maximum modulus theoremExercisesChapter 13: Approximation by Rational FunctionsPreparationRunge's theoremThe Mittag-Leffler theoremSimply connected regionsExercisesChapter 14: Conformal MappingPreservation of anglesLinear fractional transformationsNormal familiesThe Riemann mapping theoremThe class LContinuity at the boundaryConformal mapping of an annulusExercisesChapter 15: Zeros of Holomorphic FunctionsInfinite ProductsThe Weierstrass factorization theoremAn interpolation problemJensen's formulaBlaschke productsThe Muntz-Szas theoremExercisesChapter 16: Analytic ContinuationRegular points and singular pointsContinuation along curvesThe monodromy theoremConstruction of a modular functionThe Picard theoremExercisesChapter 17: Hp-SpacesSubharmonic functionsThe spaces Hp and NThe theorem of F. and M. RieszFactorization theoremsThe shift operatorConjugate functionsExercisesChapter 18: Elementary Theory of Banach AlgebrasIntroductionThe invertible elementsIdeals and homomorphismsApplicationsExercisesChapter 19: Holomorphic Fourier TransformsIntroductionTwo theorems of Paley and WienerQuasi-analytic classesThe Denjoy-Carleman theoremExercisesChapter 20: Uniform Approximation by PolynomialsIntroductionSome lemmasMergelyan's theoremExercisesAppendix: Hausdorff's Maximality TheoremNotes and CommentsBibliographyList of Special SymbolsIndex

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