Functional Analysis in Applied Mathematics and Engineering

by Pedersen; Michael

• ISBN13: 9780849371691
• ISBN10: 0849371694
• Edition: 1st
• Format: Hardcover
• Copyright: 1999-09-29
• Publisher: CRC Press

Summary

Presenting excellent material for a first course on functional analysis , Functional Analysis in Applied Mathematics and Engineering concentrates on material that will be useful to control engineers from the disciplines of electrical, mechanical, and aerospace engineering.This text/reference discusses:· rudimentary topology· Banach's fixed point theorem with applications· L^p-spaces· density theorems for testfunctions· infinite dimensional spaces· bounded linear operators· Fourier series· open mapping and closed graph theorems· compact and differential operators· Hilbert-Schmidt operators· Volterra equations· Sobolev spaces· control theory and variational analysis· Hilbert Uniqueness Method· boundary element methodsFunctional Analysis in Applied Mathematics and Engineering begins with an introduction to the important, abstract basic function spaces and operators with mathematical rigor, then studies problems in the Hilbert space setting. The author proves the spectral theorem for unbounded operators with compact inverses and goes on to present the abstract evolution semigroup theory for time dependent linear partial differential operators. This structure establishes a firm foundation for the more advanced topics discussed later in the text.

Table of Contents

Topological and Metric Spaces	1	(10)
Some Topology	1	(1)
Metric Spaces	2	(9)
Banach Spaces	11	(12)
Normed Vector Spaces	11	(1)
Lp-spaces	12	(7)
Infinite Dimensional Spaces	19	(4)
Bounded Operators	23	(10)
Basic Properties	23	(3)
Bounded Linear Operators	26	(7)
Hilbert Spaces.	33	(22)
Inner Product Spaces	33	(3)
Hilbert Spaces	36	(6)
Construction of Hilbert Spaces	42	(7)
Orthogonal Projection and Complement	49	(3)
Weak Convergence	52	(3)
Operators on Hilbert Spaces	55	(22)
The Adjoint of a Bounded Operator	55	(6)
Compactness and Compact Operators	61	(4)
Closed Operators	65	(8)
The Adjoint of an Unbounded Operator	73	(4)
Spectral Theory	77	(14)
The Spectrum and the Resolvent	78	(8)
Operator-Valued Functions	86	(5)
Integral Operators	91	(10)
Introduction	91	(1)
The Class of Hilbert-Schmidt Operators	92	(6)
Integral Equations	98	(3)
Semigroups of Evolution	101	(10)
Strongly Continuous Semigroups	101	(7)
The Resolvent	108	(3)
Sobolev Spaces	111	(16)
Basic Definitions	111	(4)
Density Theorems	115	(1)
Extension Theorems	116	(1)
Imbedding Theorems	117	(3)
Example	118	(1)
Applications	119	(1)
Example	119	(1)
The Trade Theorem	120	(4)
Negative Sobolev Spaces and Duality	124	(3)
Interpolation Spaces	127	(14)
Intermediate and Interpolation Spaces	127	(2)
The Operator L	128	(1)
Intermediate Derivatives Theorem	129	(4)
An Example	131	(1)
A Continuity Property	132	(1)
Interpolation Theorem	133	(4)
Reiteration Properties	135	(1)
Duality	136	(1)
Interpolation	137	(1)
Interpolation with Hilbert Range	138	(3)
The Space Lp(0, T; X)	138	(1)
Distributions on ]0, T[ with Values in X	139	(1)
Interpolation between Lp (0, T; H) Spaces	139	(2)
Linear Elliptic Operators	141	(32)
Elliptic Operators	142	(1)
The Dirichlet Problem, Types of Solutions	143	(2)
Boundary Operators	145	(4)
Elliptic Equations	146	(1)
The Formal Adjoint of A	147	(1)
A Modified Version of Green's Formula	148	(1)
V-elliptic and V-coercive Forms	149	(4)
The Boundary Conditions	153	(3)
Realization of the Operator A	153	(2)
The V-space	155	(1)
Variational Formulation	156	(2)
Assumptions for the Variational Problem	158	(1)
A Classical Regularity Result	159	(10)
A Regularity Theorem	159	(9)
An Abstract Regularity Theorem	168	(1)
Transposition	169	(4)
Transposition of Nonhomogeneous Boundary Value Problems	171	(2)
Regularity of Hyperbolic Mixed Problems	173	(36)
Solvability Results	174	(8)
Classical Solvability Results	174	(4)
Newer Solvability Theorems	178	(2)
A Discussion of the Regularity Results	180	(2)
Proofs of the Newer Regularity Theorems	182	(16)
Proofs of Theorems from 12.1.2	191	(7)
Interpolation Results	198	(2)
Some Additional Regularity Theorems	200	(1)
Systems with Variable Coefficients	201	(8)
A Regularity Theorem	201	(8)
The Hilbert Uniqueness Method	209	(35)
Dirichlet Boundary Control	210	(17)
Application of HUM	210	(3)
Preliminary Lemmas	213	(10)
A Discussion of the Control Area	223	(1)
Exact Controllability in Polygons and Polyhedra	224	(3)
The Hilbert Uniqueness Method	227	(6)
The Problem of Exact Controllability	228	(1)
The Operator Λ	229	(4)
The Variable Coefficients Case	233	(11)
Notation and Properties of the Operator A	233	(1)
Preliminary Lemmas	234	(2)
Application of HUM	236	(8)
Exercises	244	(47)
References	291