Contents. 2 Sequences and Series Approximation by Rational Numbers Sequences Basics on Sequences...

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1 Contents 1 Real Numbers: The Basics Notation Natural Numbers Integers Fractions and Rational Numbers Introduction Powers and Radicals of Rational Numbers Base Representation The Expansion of a Natural Number in Base b The Expansion of a Rational Number in Base b Real Numbers The Definition of a Real Number The Expansion of a Real Number in Base b The Extended Real Number System, Intervals Order Properties and the Completeness of R Cardinality of Sets Basics on Cardinality Cardinality of Z and Q Cardinality of R Cardinality of the Set of Real Functions Topology of R Introduction. Open and Closed Sets Neighborhoods, Closure, Interior Topology on a Subset Compactness Connectedness and Related Concepts The Baire Category Theorem in R Sequences and Series Approximation by Rational Numbers Sequences Basics on Sequences xi

2 xii Contents Two Particular Sequences: Arithmetic and Geometric Progressions More on Sequences Series Introduction General Criteria for Convergence of Series Series of Nonnegative Terms Series of Arbitrary Terms Rearrangement of Series Double Sequences and Double Series Product of Series The Euler Number e Infinite Products Measure Measure The Lebesgue Outer Measure The Class of Lebesgue Measurable Sets and the Lebesgue Measure Approximating Measurable Sets The Lebesgue Inner Measure The Cantor Ternary Set A Nonmeasurable Set Sequences of Sets Functions Functions on Real Numbers Introduction The Limit of a Function Continuous Functions Differentiable Functions Optimization and the Mean Value Theorem Algebra of Derivatives The Trigonometric Functions Finer Analysis of Continuity and Differentiability Differentiability of the Inverse Mapping Inverse Goniometric Functions Monotone Functions Measurable Functions Differentiability of Monotone Functions Functions of Bounded Variation Absolutely Continuous Functions and Lipschitz Functions Examples The Intermediate Value Property II

3 Contents xiii 5 Function Convergence Function Sequences Pointwise and Almost Everywhere Convergence Uniform Convergence Convergence in Measure Local Approximation by Polynomials Function Series Power Series The Taylor Series The Exponential and the Logarithmic Functions The Hyperbolic Functions The Trigonometric Functions The Binomial Series Metric Spaces Basics Mappings Between Metric Spaces More Examples (Continued) Tietze s Extension Theorem Complete Metric Spaces and the Completion of a Metric Space Separable Metric Spaces Polish Spaces Compactness in Metric Spaces Compact Metric Spaces Total Boundedness Continuous Mappings on Compact Spaces The Lebesgue Number of a Covering The Finite Intersection Property. Pseudocompactness The Baire Category Theorem Continued The Baire Category Theorem in the Context of Metric Spaces Some Applications of the Baire Category Theorem The Arzelà Ascoli Theorem Metric Fixed Point Theory The Banach Contraction Principle Continuity of the Fixed Point Integration The Riemann Integral Introduction The Definition of the Riemann Integral Properties of the Integral Functions Defined by Integrals

4 xiv Contents Some Applications of the Riemann Integral and the Arzelà Ascoli Theorem to the Theory of Ordinary Differential Equations Some Applications of the Riemann Integral and the Fixed Point Theory to the Theory of Ordinary Differential and Integral Equations Mean Value Theorems for the Riemann Integral Convergence Theorems for Riemann Integrable Functions Change of Variable; Integration by Parts Improper Riemann Integrals The Lebesgue Integral Introduction Step Functions Upper Functions Lebesgue Integrable Functions Convergence Theorems Measure and Integration Functions Defined by Integrals The Space L Riemann versus Lebesgue Integrability, and the Riemann Lebesgue Criterion for Riemann Integrability The Fundamental Theorem of Calculus for Lebesgue Integration Integration by Parts Parametric Lebesgue Integrals Convex Functions Basics on Convex Functions Some Fundamental Inequalities Jensen s Inequality Using the Exponential Function Using Powers of x (Minkowski s and Hölder s Inequalities) Fourier Series Introduction Some Elementary Trigonometric Identities The Fourier Series of 2π-periodic Lebesgue Integrable Functions The Riemann Lebesgue Lemma The Partial Sums of a Fourier Series and the Dirichlet Kernel Convergence of the Fourier Series Pointwise Convergence of the Fourier Series Cesàro Convergence of the Fourier Series Uniform Convergence of the Fourier Series Convergence of the Fourier Series in Mean Square Convergence of the Fourier Series The Fourier Integral

5 Contents xv 10 Basics on Descriptive Statistics Discrete Probability Introduction Random Variables Products of Discrete Probability Spaces Inequalities Distribution Functions Selected Distributions of Discrete Random Variables Continuous Random Variables and Their Distribution Functions Excursion to Functional Analysis Real Banach Spaces Spaces with a Norm (Normed Spaces, Banach Spaces) Operators I Finite-Dimensional Banach Spaces Infinite-Dimensional Banach Spaces Operators II Finite-Rank and Compact Operators Sets of Operators Three Basic Principles of Linear Analysis Extending Continuous Linear Functionals Bounded Sets of Operators Continuity of the Inverse Operator Complex Banach Spaces The Associated Real Normed Space Operators Linear Functionals Supporting Functionals and Differentiability Basic Results in the Complex Setting Spaces with an Inner Product (Pre-Hilbertian and Hilbert Spaces) Basic Hilbert Space Theory An Application to the Uniform Convergence of the Fourier Series Complements to Hilbert Spaces Spectral Theory Pointwise Topology and Product Spaces Excursion to Nonlinear Functional Analysis Variational Principles More on Differentiability of Convex and Lipschitz Functions More on Fixed Point Theorems An Application: Periodic Distributions Introduction The Basic Idea

6 xvi Contents The Basic Definitions Derivatives of Periodic Distributions Convergence in PD Fourier Analysis Concluding Remarks to Chapter Appendix The Set of Natural Numbers Integer Numbers Rational Numbers Real Numbers The Constructive Approach The Axiomatic Approach The Complex Number System Ordering and Choice. Three Fundamental Principles in Set Theory Definitions Examples Three Basic Principles Exercises Numbers Set-Theoretical Notations Natural Numbers Fractions Base Representation Real Numbers Cardinality of Sets and Ordinal Numbers Topology of R Sequences and Series Approximation by Rational Numbers Sequences Series The Euler Number e Measure The Lebesgue Outer Measure The Class of Lebesgue Measurable Sets and the Lebesgue Measure The Cantor Ternary Set A Nonmeasurable Set Sequences of Sets Functions Functions on Real Numbers Optimization and the Mean Value Theorem The Trigonometric Functions Finer Analysis of Continuity and Differentiability

7 Contents xvii Function Convergence Function Series Metric Spaces Integration The Riemann Integral Review of Some Frequently used Techniques for calculating Antiderivatives Improper Riemann Integral Notes on Vector-Valued Riemann Integration The Lebesgue Integral Convex Functions Fourier Series Basics on Descriptive Statistics Excursion to Functional Analysis Banach Spaces Operators Finite-Dimensional Spaces Infinite-Dimensional Spaces Operators II Three Principles of Linear Analysis Spaces with an Inner Product (Pre-Hilbertian and Hilbert spaces) Spectral Theory Pointwise Topology and Product Spaces Periodic Distributions References Author Index General Index Symbol Index

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