Functional analysis studies the algebraic, geometric, and topological structures of spaces and operators that underlie many classical problems. Individual functions satisfying specific equations are replaced by classes of functions and transforms that are determined by the particular problems at hand.
This book presents the basic facts of linear functional analysis as related to fundamental aspects of mathematical analysis and their applications. The exposition avoids unnecessary terminology and generality and focuses on showing how the knowledge of these structures clarifies what is essential in analytic problems.
The material in the first part of the book can be used for an introductory course on functional analysis, with an emphasis on the role of duality. The second part introduces distributions and Sobolev spaces and their applications. Convolution and the Fourier transform are shown to be useful tools for the study of partial differential equations. Fundamental solutions and Green's functions are considered and the theory is illustrated with several applications. In the last chapters, the Gelfand transform for Banach algebras is used to present the spectral theory of bounded and unbounded operators, which is then used in an introduction to the basic axioms of quantum mechanics.
The presentation is intended to be accessible to readers whose backgrounds include basic linear algebra, integration theory, and general topology. Almost 240 exercises will help the reader in better understanding the concepts employed.
This book is published in cooperation with Real Sociedad Matemática Española (RSME).
Readership: Graduate students interested in functional analysis, PDEs, analysis.
Author(s): Joan Cerdà
Series: Graduate Studies in Mathematics 116
Publisher: American Mathematical Society
Year: 2010
Language: English
Pages: xiv+330
Contents
Preface
Chapter 1 Introduction
1.1. Topological spaces
1.1.1. Topologies
1.1.2. Compact spaces.
1.1.3. Partitions of unity.
1.2. Measure and integration
1.2.1. Borel measures on a locally compact space X and positive linear forms on Cc(X)
1.2.2. Complex measures
1.3. Exercises
References for further reading
Chapter 2 Normed spaces and operators
2.1. Banach spaces
2.1.1. Topological vector spaces
2.1.2. Normed and Banach spaces.
2.1.3. The space C(K) and the Stone-Weierstrass theorem
2.2. Linear operators
2.2.1. Bounded linear operators
2.2.2. The space of bounded linear operators.
2.2.3. Neumann series.
2.3. Hilbert spaces
2.3.1. Scalar products
2.3.2. Orthogonal projections
2.3.3. Orthonormal bases.
2.4. Convolutions and summability kernels
2.4.1. Integral operators
2.4.2. Summability kernels on R^n
2.4.3. Periodic summability kernels.
2.5. The Riesz-Thorin interpolation theorem
2.6. Applications to linear differential equations
2.6.1. An initial value problem
2.6.2. A boundary value problem
2.7. Exercises
References for further reading
Chapter 3 Frechet spaces and Banach theorems
3.1. Frechet spaces
3.1.1. Locally convex spaces.
3.1.2. Frechet spaces.
3.2. Banach theorems
3.2.1. An application to the convergence problem of Fourier series
3.3. Exercises
References for further reading
Chapter 4 Duality
4.1. The dual of a Hilbert space
4.1.1. Riesz representation and Lax-Milgram theorem.
4.1.2. The adjoint.
4.2. Applications of the Riesz representation theorem
4.2.1. Radon-Nikodym theorem
4.2.2. The dual of L
4.2.3. The dual of C(K).
4.3. The Hahn-Banach theorem
4.3.1. Analytic form of Hahn-Banach theorem
4.3.2. The geometric Hahn-Banach theorem.
4.3.3. Extension properties
4.3.4. Proofs by duality: annihilators, total sets, completion, and the transpose
4.4. Spectral theory of compact operators
4.4.1. Elementary properties
4.4.2. The Riesz-Fredholm theory
4.5. Exercises
References for further reading
Chapter 5 Weak topologies
5.1. Weak convergence
5.2. Weak and weak* topologies
5.3. An application to the Dirichlet problem in the disc
5.4. Exercises
References for further reading
Chapter 6 Distributions
6.1. Test functions
6.2. The distributions
6.3. Differentiation of distributions
6.4. Convolution of distributions
6.4.1. Support of a distribution and distributions with compact support.
6.4.2. Convolution of distributions with functions
6.4.3. Convolution of distributions.
6.5. Distributional differential equations
6.5.1. Linear differential equations.
6.5.2. Fundamental solutions
6.5.3. Green's functions.
6.5.4. Green's function of the Dirichlet problem in the ball
6.6. Exercises
References for further reading
Chapter 7 Fourier transform and Sobolev spaces
7.1. The Fourier integral
7.2. The Schwartz class S
7.3. Tempered distributions
7.3.1. Fourier transform of tempered distributions.
7.3.2. Plancherel Theorem.
7.4. Fourier transform and signal theory
7.5. The Dirichlet problem in the half-space
7.5.1. The Poisson integral in the half-space.
7.5.2. The Hilbert transform
7.6. Sobolev spaces
7.6.1. The spaces W^m,p
7.6.2. The spaces H^m (R^n).
7.6.3. The spaces H^m (\Omega)
7.6.4. The spaces H(\Omega).
7.7. Applications
7.7.1. The Sturm-Liouville problem.
7.7.2. The Dirichlet problem.
7.7.3. Eigenvalues and eigenfunctions of the Laplacian
7.8. Exercises
References for further reading
Chapter 8 Banach algebras
8.1. Definition and examples
8.2. Spectrum
8.3. Commutative Banach algebras
8.3.1. Maximal ideals, characters, and the Gelfand transform
8.3.2. Algebras of bounded analytic functions
8.4. C*-algebras
8.4.1. Involutions
8.4.2. The Gelfand-Naimark theorem and functional calculus
8.5. Spectral theory of bounded normal operators
8.5.1. Functional calculus of normal operators
8.5.2. Spectral measures
8.5.3. Applications.
8.6. Exercises
References for further reading
Chapter 9 Unbounded operators in a Hilbert space
9.1. Definitions and basic properties
9.1.1. The adjoint
9.2. Unbounded self-adjoint operators
9.2.1. Self-adjoint operators
9.2.2. Essentially self-adjoint operators.
9.2.3. The Friedrichs extensions.
9.3. Spectral representation of unbounded self-adjoint operators
9.4. Unbounded operators in quantum mechanics
9.4.1. Position, momentum, and energy
9.4.2. States, observables, and Hamiltonian of a quantic system
9.4.3. The Heisenberg uncertainty principle and compatible observables
9.4.4. The harmonic oscillator.
9.5. Appendix: Proof of the spectral theorem
9.5.1. Functional calculus of a spectral measure
9.5.2. Unbounded functions of bounded normal operators
9.5.3. The Cayley transform.
9.5.4. Proof of Theorem 9.20:
9.6. Exercises
References for further reading
Hints to exercises
Chapter 1
Chapter 2
Chapter 3
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Chapter 8
Chapter 9
Bibliography
Index
