Functional analysis is a central subject of mathematics with applications in many areas of geometry, analysis, and physics. This book provides a comprehensive introduction to the field for graduate students and researchers. It begins in Chapter 1 with an introduction to the necessary foundations, including the Arzelà-Ascoli theorem, elementary Hilbert space theory, and the Baire Category Theorem. Chapter 2 develops the three fundamental principles of functional analysis (uniform boundedness, open mapping theorem, Hahn-Banach theorem) and discusses reflexive spaces and the James space. Chapter. Read more...
Abstract:
Functional analysis is a central subject of mathematics with applications in many areas of geometry, analysis, and physics. This book provides a comprehensive introduction to the field for graduate students and researchers. With 10 to 20 elaborate exercises at the end of each chapter, this book can be used as a text for a course on functional analysis for beginning graduate students. Read more...
Author(s): Bühler, Theo; Salamon, Dietmar A.
Series: AMS Graduate studies in mathematics 191
Publisher: American Mathematical Society
Year: 2018
Language: English
Pages: 482
Tags: Functional analysis.;Spectral theory (Mathematics);Semigroups of operators.;MATHEMATICS -- Calculus.;MATHEMATICS -- Mathematical Analysis.;Functional analysis -- Instructional exposition (textbooks, tutorial papers, etc.);Operator theory -- Instructional exposition (textbooks, tutorial papers, etc.)
Content: Cover
Title page
Contents
Preface
Introduction
Chapter 1. Foundations
1.1. Metric Spaces and Compact Sets
1.2. Finite-Dimensional Banach Spaces
1.3. The Dual Space
1.4. Hilbert Spaces
1.5. Banach Algebras
1.6. The Baire Category Theorem
1.7. Problems
Chapter 2. Principles of Functional Analysis
2.1. Uniform Boundedness
2.2. Open Mappings and Closed Graphs
2.3. Hahn-Banach and Convexity
2.4. Reflexive Banach Spaces
2.5. Problems
Chapter 3. The Weak and Weak* Topologies
3.1. Topological Vector Spaces
3.2. The Banach-Alaoglu Theorem
3.3. The Banach-Dieudonné Theorem. 3.4. The Eberlein-Šmulyan Theorem3.5. The Kreĭn-Milman Theorem
3.6. Ergodic Theory
3.7. Problems
Chapter 4. Fredholm Theory
4.1. The Dual Operator
4.2. Compact Operators
4.3. Fredholm Operators
4.4. Composition and Stability
4.5. Problems
Chapter 5. Spectral Theory
5.1. Complex Banach Spaces
5.2. Spectrum
5.3. Operators on Hilbert Spaces
5.4. Functional Calculus for Self-Adjoint Operators
5.5. Gelfand Spectrum and Normal Operators
5.6. Spectral Measures
5.7. Cyclic Vectors
5.8. Problems
Chapter 6. Unbounded Operators
6.1. Unbounded Operators on Banach Spaces. 6.2. The Dual of an Unbounded Operator6.3. Unbounded Operators on Hilbert Spaces
6.4. Functional Calculus and Spectral Measures
6.5. Problems
Chapter 7. Semigroups of Operators
7.1. Strongly Continuous Semigroups
7.2. The Hille-Yosida-Phillips Theorem
7.3. The Dual Semigroup
7.4. Analytic Semigroups
7.5. Banach Space Valued Measurable Functions
7.6. Inhomogeneous Equations
7.7. Problems
Appendix A. Zorn and Tychonoff
A.1. The Lemma of Zorn
A.2. Tychonoff's Theorem
Bibliography
Notation
Index
Back Cover. Introduction --
Foundations --
Principles of functional analysis --
The weak and weak* topologies --
Fredholm theory --
Spectral theory --
Unbounded operators --
Semigroups of operators.
