This textbook provides an introduction to functional analysis suitable for lecture courses to final year undergraduates or beginning graduates. Starting from the very basics of metric spaces, the book adopts a self-contained approach to Banach spaces and operator theory that covers the main topics, including the spectral theorem, the Gelfand transform, and Banach algebras. Various applications, such as least squares approximation, inverse problems, and Tikhonov regularization, illustrate the theory. Over 1000 worked examples and exercises of varying difficulty present the reader with ample material for reflection. This new edition of Functional Analysis has been completely revised and corrected, with many passages rewritten for clarity, numerous arguments simplified, and a good amount of new material added, including new examples and exercises. The prerequisites, however, remain the same with only knowledge of linear algebra and real analysis of a single variable assumed of the reader.
Author(s): Muscat, Joseph
Publisher: Springer
Year: 2024
Language: English
Pages: 464
Preface
Aim of This Book
Audience
Acknowledgments
Contents
1 Introduction
1.1 Preliminaries
Part I Metric Spaces
2 Distance
2.1 Balls and Open Sets
Open Sets
Relatively Open Sets
2.2 Closed Sets
Limit Points and Dense Subsets
3 Convergence and Continuity
3.1 Convergence
3.2 Continuity
Homeomorphisms
4 Completeness and Separability
4.1 Completeness
4.2 Uniformly Continuous Maps
4.3 Separable Spaces
5 Connectedness
5.1 Connected Sets
5.2 Components
6 Compactness
6.1 Bounded Sets
6.2 Totally Bounded Sets
6.3 Compact Sets
6.4 The Space C(X,Y)
Part II Banach and Hilbert Spaces
7 Normed Spaces
7.1 Vector Spaces
7.2 Norms
7.3 Metric and Vector Properties
Connected and Compact Subsets
7.4 Complete and Separable Normed Vector Spaces
7.5 Series
Convergence Tests
8 Continuous Linear Maps
8.1 Operators
8.2 Operator Norms
Matrix Norms
8.3 Isomorphisms and Projections
Projections
8.4 Quotient Spaces
8.5 Rn and Totally Bounded Sets
9 The Classical Spaces
9.1 Sequence Spaces
The Space ∞
The Space 1
The Space 2
The Space p
9.2 Function Spaces
Lebesgue Measure on Rn
Measurable Functions
Integrable Functions
Integral Operators
Approximation of Functions
The Fourier Series
10 Hilbert Spaces
10.1 Inner Products
10.2 Least Squares Approximation
Least Squares Approximation
10.3 Duality H*H
The Adjoint Map T*
10.4 Inverse Problems
10.5 Orthonormal Bases
Fourier Expansion
Examples of Orthonormal Bases
L2[a,b]—Fourier Series
L2[-1,1]—Legendre Polynomials
L2[0,∞[—Laguerre Functions
L2(R)—Hermite Functions
Applications of Orthonormal Bases
11 Banach Spaces
11.1 The Open Mapping Theorem
Complementary Subspaces
11.2 Compact Operators
Fredholm Operators
11.3 The Dual Space X*
Annihilators
The Double Dual X**
11.4 The Adjoint T
11.5 Strong and Weak Convergence
Weak Convergence
Weak Convergence in p
12 Differentiation and Integration
12.1 Differentiation
12.2 Integration for Vector-Valued Functions
Application: The Newton-Raphson Algorithm
12.3 Complex Differentiation and Integration
Part III Banach Algebras
13 Banach Algebras
13.1 Introduction
Subalgebras and Ideals
Morphisms
Representation in B(X)
13.2 Power Series
The Exponential and Logarithm Maps
13.3 The Group of Invertible Elements
13.4 Analytic Functions
14 Spectral Theory
14.1 The Spectrum of T
The Spectral Radius
14.2 The Spectrum of an Operator
The Spectrum of the Adjoint
14.3 Spectra of Compact Operators
Ascents and Descents
The Spectrum of a Compact Operator
14.4 The Functional Calculus
14.5 The Gelfand Transform
Quasinilpotents and the Radical
The State Space
The Gelfand Transform
15 C*-Algebras
15.1 Normal Elements
15.2 Normal Operators in B(H)
15.3 The Numerical Range
15.4 The Spectral Theorem for Compact Normal Operators
The Singular Value Decomposition
Application: Feature Extraction
15.5 Ideals of Compact Operators
Hilbert-Schmidt Operators
15.6 Representation Theorems
15.7 Positive Self-Adjoint Elements
Polar Decomposition
15.8 Spectral Theorem for Normal Operators
Embedding in B(H)
Hints to Selected Problems
Glossary of Symbols
Further Reading
More Advanced Books
Other References
Selected Historical Articles
Index
