Functional analysis : fundamentals and applications /
Willem, Michel,
Functional analysis : fundamentals and applications / Michel Willem. - Second edition. - Cham : Springer International Publishing AG, 2023. - 259 p. : illustrations (black and white). - Cornerstones. . - Cornerstones (Birkhäuser Verlag) .
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
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.
9783031091490
Functional analysis.
515.7 / WIL
Functional analysis : fundamentals and applications / Michel Willem. - Second edition. - Cham : Springer International Publishing AG, 2023. - 259 p. : illustrations (black and white). - Cornerstones. . - Cornerstones (Birkhäuser Verlag) .
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
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.
9783031091490
Functional analysis.
515.7 / WIL