Constructive Real Analysis

by Allen A. Goldstein

Overview

This text introduces the methods of applied functional analysis and applied convexity. Suitable for advanced undergraduates and graduate students of mathematics, science, and technology, it focuses on the solutions to two closely related problems. The first concerns finding roots of systems of equations and operative equations in a given region. The second involves extremal problems of minimizing or maximizing functions defined on subsets of finite and infinite dimensional spaces. Rather than citing practical algorithms for solving problems, this treatment provides the tools for studying problem-related algorithms.
Topics include iterations and fixed points, metric spaces, nonlinear programming, polyhedral convex programming, and infinite convex programming. Additional subjects include linear spaces and convex sets and applications to integral equations. Students should be familiar with advanced calculus and linear algebra. As an introduction to elementary functional analysis motivated by application, this volume also constitutes a helpful reference for theoretically minded engineers, scientists, and applied mathematicians.

Product Details

ISBN-13:	9780486488790
Publisher:	Dover Publications
Publication date:	05/17/2012
Series:	Dover Books on Mathematics Series
Edition description:	Reprint
Pages:	192
Product dimensions:	6.10(w) x 9.20(h) x 0.50(d)

Table of Contents

Preface xi

Chapter I Roots and Extremal Problems

Introduction 1

Section A Iterations and Fixed Points 4

A-1 Functional Iteration and Roots: One Variable 4

A-2 Newton's Method 7

A-3 Subcontractors 8

Section B Metric Spaces 10

B-1 Definitions 10

B-2 Review 11

B-3 More Definitions and Information from Analysis 12

B-4 Contraction Mapping Theorem 15

B-5 Subcontraction Mapping Theorem 17

Section C Miscellany 18

C-1 Definitions 18

C-2 Norms 19

C-3 Generalized Mean-Value Theorem 20

C-4 Spectral Bounds 21

C-5 Minimization of Some Functions 24

Section D Gradient Techniques 26

D-1 Heuristic Remarks 26

D-2 Gradient Method 27

D-3 Steepest Descent 30

D-4 Acceleration 36

D-5 Semicontinuity 40

D-6 Roots of Systems of Equations 41

D-7 Application to Linear Approximation 48

Chapter II Constraints

Section A Nonlinear Programming 54

A-1 Constraints and Penalty Functions 54

A-2 Extrema on Spheres and Supporting Hyperplanes for Convex Sets 60

Section B Polyhedral Convex Programming 67

B-1 On Homogeneous Linear Inequalities 68

B-2 Polyhedral Convex Programming 70

B-3 Implementation of the Algorithm 77

Section C Infinite Convex Programming 82

C-1 Nonpolyhedral Convex Programming I 82

C-2 Nonpolyhedral Convex Programming II 85

Chapter III Infinite Dimensional Problems

Section A Linear Spaces and Convex Sets 93

A-1 Linear Spaces 93

A-2 Normed Linear Spaces 94

A-3 Hilbert Space 97

A-4 Convex Sets in Hilbert Space 98

A-5 Projection Operator for Convex Sets 100

A-6 Distance Between Polytopes 102

A-7 On Linear Inequalities 103

Section B Miscellany 105

B-1 Linear Operators 105

B-2 Application to Mechanical Quadrature 109

B-3 The Conjugate of a Hilbert Space 115

B-4 The Frechet Differential 116

B-5 The Gateaux Differential 117

B-6 The "Chain Rule" 118

B-7 Taylor's Formula for Twice G-Differentiable Real-Valued Functions 119

B-8 Weak Convergence 120

B-9 Weak Compactness Theorem 122

B-10 Characterization of Extremals in Convex Programming 123

B-11 Convex Programming 125

B-12 Rate of Convergence 128

Section C Roots and Extremals 132

C-1 Theorem 1 (Hahn-Banach) 132

C-2 Mean-Value Theorem 137

C-3 Reflexive Spaces, Locally Uniformly Convex Spaces, and Inverse Operators 138

C-4 Newton's Method 143

C-5 Minimizing Functionals on NLS 150

Section D Applications to Integral Equations 155

D-1 Resolvent Kernel 155

D-2 Solution by Gradient Method 157

D-3 Nonlinear Integral Equations 159

Section E An Application to Control Theory 162

E-1 Rendezvous Problem 162

E-2 Application of Convex Programming 167

Notes and Bibliographic Material 169

Index 176