TY - BOOK AU - Willem,Michel AU - TI - Functional analysis: fundamentals and applications T2 - Cornerstones SN - 9783031091490 U1 - 515.7 23/eng/20230201 PY - 2023/// CY - Cham PB - Springer International Publishing AG, KW - Functional analysis N1 - Previous edition: New York: Birkhäuser, 2013; Includes bibliographical references and indexes; Intro Preface to the Second Edition Preface to the First Edition Acknowledgments Contents 1 Distance 1.1 Real Numbers 1.2 Metric Spaces 1.3 Continuity 1.4 Convergence 1.5 Comments 1.6 Exercises for Chap.1 2 The Integral 2.1 The Cauchy Integral 2.2 The Lebesgue Integral 2.3 Multiple Integrals 2.4 Change of Variables 2.5 Comments 2.6 Exercises for Chap.2 3 Norms 3.1 Banach Spaces 3.2 Continuous Linear Mappings 3.3 Hilbert Spaces 3.4 Spectral Theory 3.5 Comments 3.6 Exercises for Chap.3 4 Lebesgue Spaces 4.1 Convexity 4.2 Lebesgue Spaces 4.3 Regularization 4.4 Compactness 4.5 Comments 4.6 Exercises for Chap.4 5 Duality 5.1 Weak Convergence 5.2 James Representation Theorem 5.3 Duality of Hilbert Spaces 5.4 Duality of Lebesgue Spaces 5.5 Comments 5.6 Exercises for Chap.5 6 Sobolev Spaces 6.1 Weak Derivatives 6.2 Cylindrical Domains 6.3 Smooth Domains 6.4 Embeddings 6.5 Comments 6.6 Exercises for Chap.6 7 Capacity 7.1 Capacity 7.2 Variational Capacity 7.3 Functions of Bounded Variations 7.4 Perimeter 7.5 Distribution Theory 7.6 Comments 7.7 Exercises for Chap.7 8 Elliptic Problems 8.1 The Laplacian 8.2 Eigenfunctions 8.3 Symmetrization 8.4 Elementary Solutions 8.5 Comments 8.6 Exercises for Chap.8 9 Appendix: Topics in Calculus 9.1 Change of Variables 9.2 Surface Integrals 9.3 The Morse-Sard Theorem 9.4 The Divergence Theorem 9.5 Comments 10 Epilogue: Historical Notes on Functional Analysis 10.1 Integral Calculus 10.2 Measure and Integral 10.3 Differential Calculus 10.4 Comments References Index of Notation Index N2 - This textbook presents the principles of functional analysis in a clear and concise way. Fundamental examples are provided in the three chapters that follow: Lebesgue spaces, dual spaces, and Sobolev spaces UR - https://ezproxy.lib.gla.ac.uk/login?url=https://link.springer.com/10.1007/978-3-031-09149-0 ER -