Analysis: From Concepts to Applications

Description

This textbook covers the main results and methods of real analysis in a single volume. Taking a progressive approach to equations and transformations, this book starts with the very foundations of real analysis (set theory, order, convergence, and measure theory) before presenting powerful results that can be applied to concrete problems. In addition to classical results of functional analysis, differential calculus and integration, Analysis discusses topics such as convex analysis, dissipative operators and semigroups which are often absent from classical treatises. Acknowledging that analysis has significantly contributed to the understanding and development of the present world, the book further elaborates on techniques which pervade modern civilization, including wavelets in information theory, the Radon transform in medical imaging and partial differential equations in various mechanical and physical phenomena. Advanced undergraduate and graduate students, engineers as well as practitioners wishing to familiarise themselves with concepts and applications of analysis will find this book useful. With its content split into several topics of interest, the book’s style and layout make it suitable for use in several courses, while its self-contained character makes it appropriate for self-study. Preface Gallery Notation Contents 1 Sets, Orders, Relations and Measures 1.1 Sets and Orders Exercises 1.2 Convergence and Summability in R Exercises 1.3 Maps and Multimaps (Relations) Exercises 1.4 Measurable Spaces Exercises 1.5 Measures Exercises 1.6 Completion of a Measure Exercises 1.7 Lebesgue and Stieltjes Measures Exercises 1.8 * Product Measures Exercises 1.9 * Regular Measures on Metric Spaces Exercises Notes, Remarks, and Additional Reading 2 Encounters With Limits 2.1 Convergences Exercises 2.2 Topologies 2.2.1 General Facts About Topologies Exercises 2.2.2 Connectedness Exercises 2.2.3 Lower Semicontinuity Exercises 2.2.4 Compactness Exercises 2.3 Metric Spaces 2.3.1 General Facts About Metric Spaces Exercises 2.3.2 Complete Metric Spaces 2.3.3 Application to Ordinary Differential Equations Exercises 2.3.4 Compact Metric Spaces Exercises Additional Reading 3 Elements of Functional Analysis 3.1 Normed Spaces 3.1.1 General Properties of Normed Spaces Exercises 3.1.2 Continuity of Linear and Multilinear Maps Exercises 3.1.3 Finite Dimensional Normed Spaces Exercises 3.1.4 Series and Summable Families Exercises 3.1.5 Spaces of Continuous Functions Exercises 3.2 Topological Vector Spaces. Weak Topologies Exercises 3.3 Separation and Extension. Polarity 3.3.1 Convex Sets and Convex Functions Exercises 3.3.2 Separation and Extension Theorems Exercises 3.3.3 Polarity and Orthogonality Exercises 3.4 Couplings and Reflexivity 3.4.1 Couplings 3.4.2 Reflexivity and Weak Topologies Exercises 3.4.3 Uniform Convexity Exercises 3.4.4 Separability Exercises 3.5 Some Key Results of Functional Analysis 3.5.1 Some Classical Theorems Exercises 3.5.2 Densely Defined Operators and Transposition Exercises 3.5.3 The Spectrum of a Linear Operator Exercises 3.5.4 Compact Operators Exercises 3.6 Elementary Integration Theory 3.6.1 Regulated Functions and Their Integrals 3.6.2 *Functions of Bounded Variation and Integration Exercises 3.6.3 * Application: The Dual of C(T) Additional Reading 4 Hilbert Spaces 4.1 Hermitian Forms Exercises 4.2 Best Approximation Exercises 4.3 Orthogonal Families Exercises 4.4 The Dual of a Hilbert Space Exercises 4.5 Fourier Series 4.5.1 Application: The Dirichlet Problem for the Disk 4.5.2 Application: Dido's Problem Exercises 4.6 Orthogonal Polynomials Exercises 4.7 Elementary Spectral Theory for Self-Adjoint Operators Exercises Additional Reading 5 The Power of Differential Calculus 5.1 Differentiation of One-Variable Functions 5.1.1 Derivatives of One-Variable Functions 5.1.2 The Mean Value Theorem Exercises 5.2 Primitives and Integrals Exercises 5.3 Directional Differential Calculus Exercises 5.4 Classical Differential Calculus 5.4.1 The Main Concepts and Results of DifferentialCalculus Exercises 5.4.2 Higher Order Derivatives Exercises 5.4.3 Taylor's Formulas Exercises 5.4.4 Differentiable Partitions of Unity 5.5 Solving Equations and Inverting Maps 5.5.1 Newton's Method Exercises 5.5.2 The Inverse Mapping Theorem 5.5.3 The Implicit Function Theorem Exercises 5.5.4 Geometric Applications Exercises 5.5.5 *The Eikonal Equation 5.5.6 *Critical Points Exercises 5.5.7 *The Method of Characteristics Exercises 5.6 Applications to Optimization 5.6.1 Unconstrained Minimization Exercises 5.6.2 Normal Cones, Tangent Cones, and Constraints Exercises 5.6.3 Calculus of Tangent and Normal Cones 5.6.4 Multiplier Rules Exercises 5.7 Introduction to the Calculus of Variations 5.7.1 The One-Variable Case 5.7.2 Some Examples 5.7.3 The Legendre Transform 5.7.4 The Hamiltonian Formalism Exercises 5.7.5 The Several Variables Case Additional Reading 6 A Touch of Convex Analysis 6.1 Continuity Properties of Convex Functions Exercises 6.2 Differentiability Properties of Convex Functions 6.2.1 Derivatives of Convex Functions 6.2.2 Subdifferentials of Convex Functions Exercises 6.2.3 Differentiability of Convex Functions Exercises 6.2.4 Elementary Calculus Rules for Subdifferentials Exercises 6.2.5 Application to Optimality Conditions Exercises 6.3 The Legendre-Fenchel Transform and Its Applications 6.3.1 The Legendre-Fenchel Transform Exercises 6.3.2 A Brief Account of Convex Duality Theories Exercises 6.3.3 Duality and Subdifferentiability Results Exercises 6.3.4 The Interplay Between a Function and Its Conjugate Exercises 6.3.5 Conditioning and Well-Posedness Exercises 6.4 *Applications to the Geometry of Normed Spaces Exercises 6.5 Regularization of Convex Functions Additional Reading 7 Integration 7.1 Step Functions and μ-Measurable Functions 7.2 Integrable Functions and Their Integrals Exercises 7.3 Approximation of Integrable Functions Exercises 7.4 Convergence Results Exercises 7.5 Integrals Depending on a Parameter 7.6 Integration on a Product Exercises 7.7 Change of Variables Exercises 7.8 Measures on Spheres Exercises Additional Reading 8 Differentiation and Integration 8.1 Vectorial Measures Exercises 8.2 Decomposition and Differentiation of Measures 8.2.1 Decompositions of Measures 8.2.2 Differentiation of Measures Exercises 8.3 Differentiation of Measures on Rd Exercises 8.4 Derivatives of One-Variable Functions Exercises 8.5 Lebesgue Lp(S,E) Spaces 8.5.1 Basic Facts About Lebesgue Spaces Exercises 8.5.2 Nemytskii Maps 8.6 Duality and Reflexivity of Lebesgue Spaces Exercises 8.7 Compactness in Lebesgue Spaces Exercises 8.8 Convolution and Regularization Exercises 8.9 Some Useful Transforms 8.9.1 The Fourier Transform Exercises 8.9.2 Introduction to the Radon Transform Exercises Additional Reading 9 Partial Differential Equations 9.1 Definition and Basic Properties of Sobolev Spaces 9.1.1 Test Functions and Weak Derivatives 9.1.2 Definition and First Properties of Sobolev Spaces 9.1.3 Calculus Rules in Sobolev Spaces 9.1.4 Extension 9.1.5 Traces Exercises 9.2 Embedding Results 9.3 Elliptic Problems 9.3.1 Ellipticity 9.3.2 Energy Estimates and Existence Results 9.3.3 Regularity of Solutions 9.3.4 Maximum Principles Exercises 9.4 Nonlinear Problems 9.4.1 Transforming Equations Exercises 9.4.2 Using Potential Functions 9.4.3 Order Methods 9.4.4 Monotone Multimaps Exercises 9.4.5 Representation of Monotone Multimaps 9.4.6 Surjectivity of Maximally Monotone Multimaps Exercises 9.4.7 Sums of Maximally Monotone Multimaps Exercises 9.4.8 Variational Inequalities Additional Reading 10 Evolution Problems 10.1 Ordinary Differential Equations 10.1.1 Separation of Variables 10.1.2 Existence Results Exercises 10.1.3 Uniqueness and Globalization of Solutions Exercises 10.1.4 The Exponential Map Exercises 10.1.5 The Laplace Transform Exercises 10.2 Semigroups 10.2.1 Continuous Linear Semigroups and Their Generators Exercises 10.2.2 Characterization of Generators of Continuous Semigroups Exercises 10.2.3 * Dissipative and Accretive Multimaps Exercises 10.3 Parabolic Problems: The Heat Equation 10.4 Hyperbolic Problems: The Wave Equation Exercises Appendix: The Brouwer's Fixed Point Theorem Additional Reading References Index