Functional Analysis - Fundamentals and Applications

This textbook presents the principles of functional analysis in a clear and concise way. The first three chapters describe the general notions of distance, integral, and norm, as well as their relations. Fundamental examples are provided in the three chapters that follow: Lebesgue spaces, dual spaces, and Sobolev spaces. Two subsequent chapters develop applications to capacity theory and elliptic problems. In particular, the isoperimetric inequality and the Pólya-Szegő and Faber-Krahn inequalities are proved by purely functional methods. The epilogue contains a sketch of the history of functional analysis in relation to integration and differentiation. Starting from elementary analysis and introducing relevant research, this work is an excellent resource for students in mathematics and applied mathematics. The second edition of Functional Analysis includes several improvements as well as the addition of supplementary material. Specifically, the coverage of advanced calculus and distribution theory has been completely rewritten and expanded. New proofs, theorems, and applications have been added as well for readers to explore. Preface to the Second Edition 7 Preface to the First Edition 8 Acknowledgments 11 Contents 12 1 Distance 15 1.1 Real Numbers 15 1.2 Metric Spaces 18 1.3 Continuity 23 1.4 Convergence 29 1.5 Comments 33 1.6 Exercises for Chap.1 34 2 The Integral 37 2.1 The Cauchy Integral 37 2.2 The Lebesgue Integral 40 2.3 Multiple Integrals 57 2.4 Change of Variables 62 2.5 Comments 66 2.6 Exercises for Chap.2 67 3 Norms 72 3.1 Banach Spaces 72 3.2 Continuous Linear Mappings 80 3.3 Hilbert Spaces 89 3.4 Spectral Theory 95 3.5 Comments 98 3.6 Exercises for Chap.3 99 4 Lebesgue Spaces 101 4.1 Convexity 101 4.2 Lebesgue Spaces 106 4.3 Regularization 111 4.4 Compactness 118 4.5 Comments 120 4.6 Exercises for Chap.4 121 5 Duality 125 5.1 Weak Convergence 125 5.2 James Representation Theorem 129 5.3 Duality of Hilbert Spaces 131 5.4 Duality of Lebesgue Spaces 137 5.5 Comments 141 5.6 Exercises for Chap.5 141 6 Sobolev Spaces 143 6.1 Weak Derivatives 143 6.2 Cylindrical Domains 153 6.3 Smooth Domains 157 6.4 Embeddings 160 6.5 Comments 170 6.6 Exercises for Chap.6 171 7 Capacity 176 7.1 Capacity 176 7.2 Variational Capacity 179 7.3 Functions of Bounded Variations 185 7.4 Perimeter 189 7.5 Distribution Theory 193 7.6 Comments 204 7.7 Exercises for Chap.7 204 8 Elliptic Problems 207 8.1 The Laplacian 207 8.2 Eigenfunctions 210 8.3 Symmetrization 215 8.4 Elementary Solutions 225 8.5 Comments 227 8.6 Exercises for Chap.8 227 9 Appendix: Topics in Calculus 230 9.1 Change of Variables 230 9.2 Surface Integrals 232 9.3 The Morse–Sard Theorem 235 9.4 The Divergence Theorem 237 9.5 Comments 239 10 Epilogue: Historical Notes on Functional Analysis 240 10.1 Integral Calculus 240 10.2 Measure and Integral 244 10.3 Differential Calculus 247 10.4 Comments 250 References 251 Index of Notation 255 Index 257 This textbook presents the principles of functional analysis in a clear and concise way. The first three chapters describe the general notions of distance, integral, and norm, as well as their relations. Fundamental examples are provided in the three chapters that follow: Lebesgue spaces, dual spaces, and Sobolev spaces. Two subsequent chapters develop applications to capacity theory and elliptic problems. In particular, the isoperimetric inequality and the Polya-Szego and Faber-Krahn inequalities are proved by purely functional methods. The epilogue contains a sketch of the history of functional analysis in relation to integration and differentiation. Starting from elementary analysis and introducing relevant research, this work is an excellent resource for students in mathematics and applied mathematics. The second edition of Functional Analysis includes several improvements as well as the addition of supplementary material. Specifically, the coverage of advanced calculus and distribution theory has been completely rewritten and expanded. New proofs, theorems, and applications have been added as well for readers to explore