 Linear Functional Analysis

Graduate Studies in Mathematics, American Mathematical Society and Real Sociedad Matemática Española, 2010

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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 can be used for an introductory course on functional analysis, with an emphasis on the role of duality. The second part introduces distributions, 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 background includes basic linear algebra, integration theory and general topology. Almost 230 exercises will help the reader in better understanding of concepts employed.

Contents:
Chapter 1. Introduction: topological spaces and measure theory.
Chapter 2. Normed spaces and operators: Banach spaces, linear operators, Hilbert spaces, convolutions and summability kernels, the Riesz-Thorin interpolation theorem, applications to linear differential equations.
Chapter 3. Fréchet spaces and Banach theorems: Fréchet spaces, Banach theorems.
Chapter 4. Duality: the dual of a Hilbert space, applications of the Riesz representation theorem, the Hahn-Banach theorem, spectral theory of compact operators.
Chapter 5. Weak topologies: weak convergence, weak and weak-star topologies, an application to the Dirichlet problem in the disc.
Chapter 6. Distributions: test functions, the distributions, differentiation of distributions, convolution of distributions, distributional differential equations.
Chapter 7. Fourier transform and Sobolev spaces: the Fourier integral, the Schwartz class, tempered distributions, Fourier transform and signal theory, the Dirichlet problem in the half-plane, Sobolev spaces, applications.
Chapter 8. Banach algebras: definition and examples, spectrum, commutative Banach algebras, C star-algebras, spectral theory of bounded normal operators.
Chapter 9. Unbounded operators in a Hilbert space: definitions and basic properties, unbounded self-adjoint operators, spectral representation of unbounded self-adjoint operators, unbounded operators and quantum mechanics, appendix: proof of the spectral theorem.

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