Functional Analysis. Fundamentals and Applications

✍ Scribed by Michel Willem

• Publisher: Birkhäuser
• Year: 2022
• Tongue: English
• Leaves: 260
• Series: Cornerstones
• Edition: 2
• Category: Library

✦ Table of Contents

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