Hardcover ISBN:  9780821851159 
Product Code:  GSM/116 
List Price:  $70.00 
MAA Member Price:  $63.00 
AMS Member Price:  $56.00 
Electronic ISBN:  9781470411787 
Product Code:  GSM/116.E 
List Price:  $66.00 
MAA Member Price:  $59.40 
AMS Member Price:  $52.80 

Book DetailsGraduate Studies in MathematicsVolume: 116; 2010; 330 ppMSC: Primary 46; Secondary 47;
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 Sociedád Matematica Española.ReadershipGraduate students interested in functional analysis, PDEs, analysis.

Table of Contents

Chapters

Chapter 1. Introduction

Chapter 2. Normed spaces and operators

Chapter 3. Fréchet spaces and Banach theorems

Chapter 4. Duality

Chapter 5. Weak topologies

Chapter 6. Distributions

Chapter 7. Fourier transform and Sobolev spaces

Chapter 8. Banach algebras

Chapter 9. Unbounded operators in a Hilbert space

Hints to exercises


Additional Material

Reviews

Overall, this is an interesting and wellwritten book, covering a lot of material. I found it very readable... The discussion is well motivated mathematically (there are few applications to other areas) and full details of proofs are given, so it is fairly selfcontained.
Bryan P. Rynne, Mathematical Reviews


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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 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.
Graduate students interested in functional analysis, PDEs, analysis.

Chapters

Chapter 1. Introduction

Chapter 2. Normed spaces and operators

Chapter 3. Fréchet spaces and Banach theorems

Chapter 4. Duality

Chapter 5. Weak topologies

Chapter 6. Distributions

Chapter 7. Fourier transform and Sobolev spaces

Chapter 8. Banach algebras

Chapter 9. Unbounded operators in a Hilbert space

Hints to exercises

Overall, this is an interesting and wellwritten book, covering a lot of material. I found it very readable... The discussion is well motivated mathematically (there are few applications to other areas) and full details of proofs are given, so it is fairly selfcontained.
Bryan P. Rynne, Mathematical Reviews