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.
Author(s): Michel Willem
Series: Cornerstones
Edition: 2
Publisher: Birkhäuser
Year: 2023
Language: English
Pages: 251
Tags: Lebesgue Integral, Banach Spaces, Hilbert Spaces, Spectral Theory, Lebesgue Spaces, Duality, Sobolev Spaces, Capacity, Elliptic Problems
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
