This textbook is addressed to graduate students in mathematics or other disciplines who wish to understand the essential concepts of functional analysis and their applications to partial differential equations.
The book is intentionally concise, presenting all the fundamental concepts and results but omitting the more specialized topics. Enough of the theory of Sobolev spaces and semigroups of linear operators is included as needed to develop significant applications to elliptic, parabolic, and hyperbolic PDEs. Throughout the book, care has been taken to explain the connections between theorems in functional analysis and familiar results of finite-dimensional linear algebra.
The main concepts and ideas used in the proofs are illustrated with a large number of figures. A rich collection of homework problems is included at the end of most chapters. The book is suitable as a text for a one-semester graduate course.
Readership: Graduate students interested in functional analysis and partial differential equations.
Author(s): Alberto Bressan
Series: Graduate Studies in Mathematics 143
Publisher: American Mathematical Society
Year: 2012
Language: English
Pages: xii+250
Preface
Chapter 1 Introduction
1.1. Linear equations
1.2. Evolution equations
1.3. Function spaces
1.4. Compactness
Chapter 2 Banach Spaces
2.1. Basic definitions
2.1.1. Examples of normed and Banach spaces.
2.2. Linear operators
2.2.1. Examples of linear operators.
2.3. Finite-dimensional spaces
2.4. Seminorms and Frechet spaces
2.5. Extension theorems
2.6. Separation of convex sets
2.7. Dual spaces and weak convergence
2.7.1. Weak convergence.
2.7.2. Weak-star convergence
2.8. Problems
Chapter 3 Spaces of Continuous Functions
3.1. Bounded continuous functions
3.2. The Stone-Weierstrass approximation theorem
3.2.1. Complex-valued functions
3.3. Ascoli's compactness theorem
3.4. Spaces of Holder continuous functions
3.5. Problems
Chapter 4 Bounded Linear Operators
4.1. The uniform boundedness principle
4.2. The open mapping theorem
4.3. The closed graph theorem
4.4. Adjoint operators
4.5. Compact operators
4.5.1. Integral operators.
4.6. Problems
Chapter 5 Hilbert Spaces
5.1. Spaces with an inner product
5.2. Orthogonal projections
5.3. Linear functionals on a Hilbert space
5.4. Gram-Schmidt orthogonalization
5.5. Orthonormal sets
5.5.1. Fourier series.
5.6. Positive definite operators
5.7. Weak convergence
5.8. Problems
Chapter 6 Compact Operators on a Hilbert Space
6.1. Fredholm theory
6.2. Spectrum of a compact operator
6.3. Selfadjoint operators
6.4. Problems
Chapter 7 Semigroups of Linear Operators
7.1. Ordinary differential equations in a Banach space
7.1.1. Linear homogeneous ODEs.
7.2. Semigroups of linear operators
7.2.1. Definition and basic properties of semigroups
7.3. Resolvents
7.4. Generation of a semigroup
7.5. Problems
Chapter 8 Sobolev Spaces
8.1. Distributions and weak derivatives
8.1.1. Distributions.
8.1.2. Weak derivatives.
8.2. Mollificat ions
8.3. Sobolev spaces
8.4. Approximations of Sobolev functions
8.5. Extension operators
8.6. Embedding theorems
8.6.1. Morrey's inequality
8.6.2. The Gagliardo-Nirenberg inequality
8.6.3. High-order Sobolev estimates.
8.7. Compact embeddings
8.8. Differentiability properties
8.9. Problems
Chapter 9 Linear Partial Differential Equations
9.1. Elliptic equations
9.1.1. Physical interpretation.
9.1.2. Classical and weak solutions
9.1.3. Homogeneous second-order elliptic operators
9.1.4. Representation of solutions in terms of eigenfunctions
9.1.5. More general linear elliptic operators.
9.2. Parabolic equations
9.2.1. Representation of solutions in terms of eigenfunctions
9.2.2. More general operators
9.3. Hyperbolic equations
9.4 Problems
Appendix Background Material
A.1. Partially ordered sets
A.2. Metric and topological spaces
A.2.1. Fixed points of contractive maps
A.2.2. The Baire category theorem
A.3. Review of Lebesgue measure theory
A.3.1. Measurable sets
A.3.2. Lebesgue integration
A.4. Integrals of functions taking values in a Banach space
A.5. Mollifications
A.5.1. Partitions of unity.
A.6. Inequalities
A.6.1. Convex sets and convex functions
A.6.2. Basic inequalities
A.6.3. A differential inequality
A.7. Problems
Summary of Notation
Bibliography
Index
