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Basic Real Analysis by Anthony W. Knapp


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Basic Real Analysis written by Anthony W. Knapp. This is an other book of mathematics cover the following topics.

  • THEORY OF CALCULUS IN ONE REAL VARIABLE
    Review of Real Numbers, Sequences, Continuity, Interchange of Limits, Uniform Convergence, Riemann Integral, Complex-Valued Functions, Taylor’s Theorem with Integral Remainder, Power Series and Special Functions, Summability, Weierstrass Approximation Theorem, Fourier Series, Problems

  • METRIC SPACES
    Definition and Examples, Open Sets and Closed Sets, Continuous Functions, Sequences and Convergence, Subspaces and Products, Properties of Metric Spaces, Compactness and Completeness, Connectedness, Baire Category Theorem, Properties of C(S) for Compact Metric S, Completion, Problems

  • THEORY OF CALCULUS IN SEVERAL REAL VARIABLES
    Operator Norm, Nonlinear Functions and Differentiation, Vector-Valued Partial Derivatives and Riemann Integrals, Exponential of a Matrix, Partitions of Unity, Inverse and Implicit Function Theorems, Definition and Properties of Riemann Integral, Riemann Integrable Functions, Fubini’s Theorem for the Riemann Integral, Change of Variables for the Riemann Integral, Arc Length and Integrals with Respect to Arc Length, Line Integrals and Conservative Vector Fields, Green’s Theorem in the Plane, Problems

  • THEORY OF ORDINARY DIFFERENTIAL EQUATIONS AND SYSTEMS
    Qualitative Features and Examples, Existence and Uniqueness, Dependence on Initial Conditions and Parameters, Integral Curves, Linear Equations and Systems, Wronskian, Homogeneous Equations with Constant Coefficients, Homogeneous Systems with Constant Coefficients, Series Solutions in the Second-Order Linear Case, Problems

  • LEBESGUE MEASURE AND ABSTRACT MEASURE THEORY
    Measures and Examples, Measurable Functions, Lebesgue Integral, Properties of the Integral, Proof of the Extension Theorem, Completion of a Measure Space, Fubini’s Theorem for the Lebesgue Integral, Integration of Complex-Valued and Vector-Valued Functions, L1, L2, L∞, and Normed Linear Spaces, Arc Length and Lebesgue Integration, Problems

  • MEASURE THEORY FOR EUCLIDEAN SPACE
    Lebesgue Measure and Other Borel Measures, Convolution, Borel Measures on Open Sets, Comparison of Riemann and Lebesgue Integrals, Change of Variables for the Lebesgue Integral, Hardy–Littlewood Maximal Theorem, Fourier Series and the Riesz–Fischer Theorem, Stieltjes Measures on the Line, Fourier Series and the Dirichlet–Jordan Theorem, Distribution Functions, Problems

  • DIFFERENTIATION OF LEBESGUE INTEGRALS ON THE LINE
    Differentiation of Monotone Functions, Absolute Continuity, Singular Measures, and Lebesgue Decomposition, Problems

  • FOURIER TRANSFORM IN EUCLIDEAN SPACE
    Elementary Properties, Fourier Transform on L1, Inversion Formula, Fourier Transform on L2, Plancherel Formula, Schwartz Space, Poisson Summation Formula, Poisson Integral Formula, Hilbert Transform, Problems

  • L p SPACES
    Inequalities and Completeness, Convolution Involving L p, Jordan and Hahn Decompositions, Radon–Nikodym Theorem, Continuous Linear Functionals on L p, Riesz–Thorin Convexity Theorem, Marcinkiewicz Interpolation Theorem, Problems

  • TOPOLOGICAL SPACES
    Open Sets and Constructions of Topologies, Properties of Topological Spaces, Compactness and Local Compactness, Product Spaces and the Tychonoff Product Theorem, Sequences and Nets, Quotient Spaces, Urysohn’s Lemma, Metrization in the Separable Case, Ascoli–Arzela` and Stone–Weierstrass Theorems, Problems

  • INTEGRATION ON LOCALLY COMPACT SPACES
    Setting, Riesz Representation Theorem, Regular Borel Measures, Dual to Space of Finite Signed Measures, Problems

  • HILBERT AND BANACH SPACES
    Definitions and Examples, Geometry of Hilbert Space, Bounded Linear Operators on Hilbert Spaces, Hahn–Banach Theorem, Uniform Boundedness Theorem, Interior Mapping Principle, Problems

  • APPENDIX A. BACKGROUND TOPICS
    Sets and Functions, Mean Value Theorem and Some Consequences, Inverse Function Theorem in One Variable, Complex Numbers, Classical Schwarz Inequality, Equivalence Relations, Linear Transformations, Matrices, and Determinants, Factorization and Roots of Polynomials, Partial Orderings and Zorn’s Lemma, Cardinality

  • APPENDIX B. ELEMENTARY COMPLEX ANALYSIS
    Complex Derivative and Analytic Functions, Complex Line Integrals, Goursat’s Lemma and the Cauchy Integral Theorem, Cauchy Integral Formula, Taylor’s Theorem, Local Properties of Analytic Functions, Logarithms and Winding Numbers, Operations on Taylor Series, Argument Principle, Residue Theorem, Evaluation of Definite Integrals, Global Theorems in Simply Connected Regions, Global Theorems in General Regions, Laurent Series, Holomorphic Functions of Several Variables, Problems

  • CONTENTS OF ADVANCED REAL ANALYSIS
    Introduction to Boundary-Value Problems, Compact Self-Adjoint Operators, Topics in Euclidean Fourier Analysis, Topics in Functional Analysis, Distributions, Compact and Locally Compact Groups, Aspects of Partial Differential Equations, Analysis on Manifolds, Foundations of Probability, Introduction to Wavelets,

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