Table of Contents
Preface vii
1 The Hahn-Banach Theorems. Introduction to the Theory of Conjugate Convex Functions 1
1.1 The Analytic Form of the Hahn-Banach Theorem: Extension of Linear Functional 1
1.2 The Geometric Forms of the Hahn-Banach Theorem: Separation of Convex Sets 4
1.3 The Bidual E**. Orthogonality Relations 8
1.4 A Quick Introduction to the Theory of Conjugate Convex Functions 10
Comments on Chapter 1 17
Exercises for Chapter 1 19
2 The Uniform Boundedness Principle and the Closed Graph Theorem 31
2.1 The Baire Category Theorem 31
2.2 The Uniform Boundedness Principle 32
2.3 The Open Mapping Theorem and the Closed Graph Theorem 34
2.4 Complementary Subspaces. Right and Left Invertibility of Linear Operators 37
2.5 Orthogonality Revisited 40
2.6 An Introduction to Unbounded Linear Operators. Definition of the Adjoint 43
2.7 A Characterization of Operators with Closed Range. A Characterization of Surjective Operators 46
Comments on Chapter 2 48
Exercises for Chapter 2 49
3 Weak Topologies. Reflexive Spaces. Separable Spaces. Uniform Convexity 55
3.1 The Coarsest Topology for Which a Collection of Maps Becomes Continuous 55
3.2 Definition and Elementary Properties of the Weak Topology σ(E, E*) 57
3.3 Weak Topology, Convex Sets, and Linear Operators 60
3.4 The Weak* Topology σ(E* E) 62
3.5 Reflexive Spaces 67
3.6 Separable Spaces 72
3.7 Uniformly Convex Spaces 76
Comments on Chapter 3 78
Exercises for Chapter 3 79
4 Lp Spaces 89
4.1 Some Results about Integration That Everyone Must Know 90
4.2 Definition and Elementary Properties of Lp Spaces 91
4.3 Reflexivity. Separability. Dual of Lp 95
4.4 Convolution and regularization 104
4.5 Criterion for Strong Compactness in Lp 111
Comments on Chapter 4 114
Exercises for Chapter 4 118
5 Hilbert Spaces 131
5.1 Definitions and Elementary Properties. Projection onto a Closed Convex Set 131
5.2 The Dual Space of a Hilbert Space 135
5.3 The Theorems of Stampacchia and Lax-Milgram 138
5.4 Hilbert Sums. Orthonormal Bases 141
Comments on Chapter 5 144
Exercises for Chapter 5 146
6 Compact Operators. Spectral Decomposition of Self-Adjoint Compact Operators 157
6.1 Definitions. Elementary Properties. Adjoint 157
6.2 The Riesz-Fredholm Theory 159
6.3 The Spectrum of a Compact Operator 162
6.4 Spectral Decomposition of Self-Adjoint Compact Operators 165
Comments on Chapter 6 168
Exercises for Chapter 6 170
7 The Hille-Yosida Theorem 181
7.1 Definition and Elementary Properties of Maximal Monotone Operators 181
7.2 Solution of the Evolution Problem du/dt + Au = 0 on (0, +∞), u(0) = u0. Existence and uniqueness 184
7.3 Regularity 191
7.4 The Self-Adjoint Case 193
Comments on Chapter 7 197
8 Sobolev Spaces and the Variational Formulation of Boundary Value Problems in One Dimension 201
8.1 Motivation 201
8.2 The Sobolev Space W1.p (I) 202
8.3 The Space &$$$; 217
8.4 Some Examples of Boundary Value Problems 220
8.5 The Maximum Principle 229
8.6 Eigenfunctions and Spectral Decomposition 231
Comments on Chapter 8 233
Exercises for Chapter 8 235
9 Sobolev Spaces and the Variational Formulation of Elliptic Boundary Value Problems in N Dimensions 263
9.1 Definition and Elementary Properties of the Sobolev Spaces W1.p (Ω) 263
9.2 Extension Operators 272
9.3 Sobolev Inequalities 278
9.4 The Space &$$$; (Ω) 287
9.5 Variational Formulation of Some Boundary Value Problems 291
9.6 Regularity of Weak Solutions 298
9.7 The Maximum Principle 307
9.8 Eigenfunctions and Spectral Decomposition 311
Comments on Chapter 9 312
10 Evolution Problems: The Heat Equation and the Wave Equation 325
10.1 The Heat Equation: Existence, Uniqueness, and Regularity 325
10.2 The Maximum Principle 333
10.3 The Wave Equation 335
Comments on Chapter 10 340
11 Miscellaneous Complements 349
11.1 Finite-Dimensional and Finite-Codimensional Spaces 349
11.2 Quotient Spaces 353
11.3 Some Classical Spaces of Sequences 357
11.4 Banach Spaces over C: What Is Similar and What Is Different? 361
Solutions of Some Exercises 371
Problems 435
Partial Solutions of the Problems 521
Notation 583
References 585
Index 595
