APPLIED FUNCTIONAL ANALYSIS

Transcription

1 APPLIED FUNCTIONAL ANALYSIS Second Edition JEAN-PIERRE AUBIN University of Paris-Dauphine Exercises by BERNARD CORNET and JEAN-MICHEL LASRY Translated by CAROLE LABROUSSE A Wiley-Interscience Publication JOHN WILEY & SONS, INC. New York Chichester Weinheim Brisbane Singapore Toronto

2 CONTENTS Preface Introduction: A Guide to the Reader 1. The Projection Theorem 1.1. Definition of a Hubert Space, Review of Continuous Linear and Bilinear Operators, Extension of Continuous Linear and Bilinear Operators by Density, The Best Approximation Theorem, Orthogonal Projectors, Closed Subspaces, Quotient Spaces, and Finite Products of Hilbert Spaces, 22 *1.7. Orthogonal Bases for a Separable Hilbert Space, Theorems on Extension and Separation 2.1. Extension of Continuous Linear and Bilinear Operators, 2.2. A Density Criterion, Separation Theorems, A Separation Theorem in Finite Dimensional Spaces, Support Functions, 32 *2.6. The Duality Theorem in Convex Optimization, 34 *2.7. Von Neumann's Minimax Theorem, 39 *2.8. Characterization of Pareto Optima, Dual Spaces and Transposed Operators 3.1. The Dual of a Hilbert Space, Realization of the Dual of a Hilbert Space, Transposition of Operators, Transposition of Injective Operators, 57

3 vüi CONTENTS 3.5. Duals of Finite Products, Quotient Spaces, and Closed or Dense Subspaces, The Theorem of Lax-Milgram, 64 *3.7. Variational Inequalities, 65 *3.8. Noncooperative Equilibria in л-person Quadratic Games, The Banach Theorem and the Banach-Steinhaus Theorem Properties of Bounded Sets of Operators, The Mean Ergodic Theorem, The Banach Theorem, The Closed Range Theorem, Characterization of Left Invertible Operators, Characterization of Right Invertible Operators, 86 *4.7. Quadratic Programming with Linear Constraints, Construction of Hubert Spaces The Initial Scalar Product, The Final Scalar Product, Normal Subspaces of a Pivot Space, Minimal and Maximal Domains of a Closed Family of Operators, 104 *5.5. Unbounded Operators and Their Adjoints, 107 *5.6. Completion of a Pre-Hilbert Space Contained in a Hubert Space, 110 *5.7. HausdorffCompletion, 111 *5.8. The Hilbert Sum of Hilbert Spaces, 112 *5.9. Reproducing Kernels of a Hilbert Space of Functions, L 1 Spaces and Convolution Operators The Space L 2 (Cl) of Square Integrable Functions, The Spaces L 2 (Q, a) with Weights, The Spaced, The Convolution Product for Functions of ^(W) and of L l (U"), Convolution Operators, Approximation by Convolution, 133 *6.7. Example. Convolution Power for Characteristic Functions, 135 *6.8. Example. Convolution Product for Polynomials: Appell Polynomials, 139

4 IX 7. Sobolev Spaces of Functions of One Variable The Space Я 0 т (П) and Its Dual H~ m (Q), Definition of Distributions, Differentiation of Distributions, Relations Between Я 0 Ш ( 2) and H^{U), The Sobolev Space Я т (П), Relations Between H m (Q) and H m (U), 158 *7.7. Characterization of the Dual of Я т (0), Trace Theorems, Convolution of Distributions, Some Approximation Procedures in Spaces of Functions Approximation by Orthogonal Polynomials, Legendre, Laguerre, and Hermite Polynomials, Fourier Series, Approximation by Step Functions, Approximation by Piecewise Polynomial Functions, Approximation in Sobolev Spaces, Sobolev Spaces of Functions of Several Variables and the Fourier Transform The Sobolev Spaces Я 0 т (а), H m ( l), and H~ m {Q), The Fourier Transform of Infinitely Differentiable and Rapidly Decreasing Functions, The Fourier Transform of Sobolev Spaces, The Trace Theorem for the Spaces Н т (Щ), The Trace Theorem for the Spaces H m (Q), The Compactness Theorem, Introduction to Set-Valued Analysis and Convex Analysis Graphical Derivations, Jump Maps of Vector Distributions, Epiderivatives, Dual Concepts, Conjugate Functions, Economic Optima, Elementary Spectral Theory Compact Operators, The Theory of Riesz-Fredholm, Characterization of Compact Operators from One Hilbert Space to Another, 266

5 X CONTENTS The Fredholm Alternative, 268 *11.5. Applications: Constructions of Intermediate Spaces, 271 *11.6. Application: Best Approximation Processes, 274 * Perturbation of an Isomorphism by a Compact Operator, Hilbert-Schmidt Operators and Tensor Products The Hilbert Space of Hilbert-Schmidt Operators, The Fundamental Isomorphism Theorem, Hilbert Tensor Products, The Tensor Product of Continuous Linear Operators, The Hilbert Tensor Product by/ 2, The Hilbert Tensor Product by L 2, The Tensor Product by the Sobolev Space H m, Boundary Value Problems The Formal Adjoint of an Operator and Green's Formula, Green's Formula for Bilinear Forms, Abstract Variational Boundary Value Problems, Examples of Boundary Value Problems, Approximation of Solutions to Neumann Problems, Restriction and Extension of the Formal Adjoint, Unilateral Boundary Value Problems, Introduction to Calculus of Variations, Differential-Operational Equations and Semigroups of Operators Semigroups of Operators, Characterization of Infinitesimal Generators of Semigroups, Differential-Operational Equations, Boundary Value Problems for Parabolic Equations, Systems Theory: Internal and External Representations, Viability Kernels and Capture Basins The Nagumo Theorem, Viability Kernels and Capture Basins, First-Order Partial Differential Equations Some Hamilton-Jacobi Equations, Systems of First-Order Partial Differential Equations, 428

6 CONTENTS XI Lotka-McKendrick Systems, Distributed Boundary Data, 445 Selection of Results General Properties, Properties of Continuous Linear Operators, Separation Theorems and Polarity, Construction of Hilbert Spaces, Compact Operators, Semigroup of Operators, The Green's Formula, Set-Valued Analysis and Optimization, Convex Analysis, Minimax Inequalities, Sobolev Spaces, Convolution, and Fourier Transform, Viability Kernels and Capture Basins, First-Order Partial Differential Equations, 467 Exercises 470 Bibliography 488 Index 493