9780444517906

Applications of Functional Analysis And Operator Theory

by ; ;
  • ISBN13:

    9780444517906

  • ISBN10:

    0444517901

  • Edition: 2nd
  • Format: Hardcover
  • Copyright: 2005-04-08
  • Publisher: Elsevier Science
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Summary

Functional analysis is a powerful tool when applied to mathematical problems arising from physical situations. The present book provides, by careful selection of material, a collection of concepts and techniques essential for the modern practitioner. Emphasis is placed on the solution of equations (including nonlinear and partial differential equations). The assumed background is limited to elementary real variable theory and finite-dimensional vector spaces. Key Features - Provides an ideal transition between introductory math courses and advanced graduate study in applied mathematics, the physical sciences, or engineering. - Gives the reader a keen understanding of applied functional analysis, building progressively from simple background material to the deepest and most significant results. - Introduces each new topic with a clear, concise explanation. - Includes numerous examples linking fundamental principles with applications. - Solidifies the readers understanding with numerous end-of-chapter problems.

Table of Contents

Preface v
Acknowledgements ix
Contents xi
Banach Spaces
1(38)
Introduction
1(2)
Vector Spaces
3(4)
Normed Vector Spaces
7(11)
Banach Spaces
18(9)
Hilbert Space
27(12)
Problems
36(3)
Lebesgue Integration and the Lp Spaces
39(26)
Introduction
39(2)
The Measure of a Set
41(7)
Measurable Functions
48(2)
Integration
50(7)
The Lp Spaces
57(3)
Applications
60(5)
Problems
62(3)
Foundations of Linear Operator Theory
65(50)
Introduction
65(2)
The Basic Terminology of Operator Theory
67(2)
Some Algebraic Properties of Linear Operators
69(5)
Continuity and Boundedness
74(8)
Some Fundamental Properties of Bounded Operators
82(8)
First Results on the Solution of the Equation Lf = g
90(6)
Introduction to Spectral Theory
96(5)
Closed Operators and Differential Equations
101(14)
Problems
108(7)
Introduction to Nonlinear Operators
115(32)
Introduction
115(2)
Preliminaries
117(4)
The Contraction Mapping Principle
121(9)
The Frechet Derivative
130(6)
Newton's Method for Nonlinear Operators
136(11)
Problems
143(4)
Compact Sets in Banach Spaces
147(10)
Introduction
147(1)
Definitions
148(2)
Some Consequences of Compactness
150(3)
Some Important Compact Sets of Functions
153(4)
Problems
155(2)
The Adjoint Operator
157(32)
Introduction
157(1)
The Dual of a Banach Space
158(7)
Weak Convergence
165(3)
Hilbert Space
168(2)
The Adjoint of a Bounded Linear Operator
170(5)
Bounded Self-adjoint Operators --- Spectral Theory
175(4)
The Adjoint of an Unbounded Linear Operator in Hilbert Space
179(10)
Problems
184(5)
Linear Compact Operators
189(28)
Introduction
189(1)
Examples of Compact Operators
190(5)
The Fredholm Alternative
195(4)
The Spectrum
199(3)
Compact Self-adjoint Operators
202(3)
The Numerical Solution of Linear Integral Equations
205(12)
Problems
212(5)
Nonlinear Compact Operators and Monotonicity
217(24)
Introduction
217(3)
The Schauder Fixed Point Theorem
220(4)
Positive and Monotone Operators in Partially Ordered Banach Spaces
224(17)
Problems
236(5)
The Spectral Theorem
241(28)
Introduction
241(2)
Preliminaries
243(7)
Background to the Spectral Theorem
250(4)
The Spectral Theorem for Bounded Self-adjoint Operators
254(4)
The Spectrum and the Resolvent
258(4)
Unbounded Self-adjoint Operators
262(2)
The Solution of an Evolution Equation
264(5)
Problems
266(3)
Generalized Eigenfunction Expansions Associated with Ordinary Differential Equations
269(34)
Introduction
269(2)
Extensions of Symmetric Operators
271(7)
Formal Ordinary Differential Operators: Preliminaries
278(2)
Symmetric Operators Associated with Formal Ordinary Differential Operators
280(5)
The Construction of Self-adjoint Extensions
285(7)
Generalized Eigenfunction Expansions
292(11)
Problems
299(4)
Linear Elliptic Partial Differential Equations
303(40)
Introduction
303(2)
Notation
305(3)
Weak Derivatives and Sobolev Spaces
308(10)
The Generalized Dirichlet Problem
318(6)
Fredholm Alternative for Generalized Dirichlet Problem
324(3)
Smoothness of Weak Solutions
327(9)
Further Developments
336(7)
Problems
338(5)
The Finite Element Method
343(16)
Introduction
343(1)
The Ritz Method
344(7)
The Rate of Convergence of the Finite Element Method
351(8)
Problems
356(3)
Introduction to Degree Theory
359(26)
Introduction
359(6)
The Degree in Finite Dimensions
365(8)
The Leray-Schauder Degree
373(4)
A Problem in Radiative Transfer
377(8)
Problems
381(4)
Bifurcation Theory
385(24)
Introduction
385(3)
Local Bifurcation Theory
388(7)
Global Eigenfunction Theory
395(14)
Problems
406(3)
References 409(8)
List of Symbols 417(4)
Index 421

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