Convex Functional Analysis (eBook)

Contents 6
List of Figures 10
Preface 12
Overview of Book 12
Organization 13
Acknowledgements 14
Chapter 1 Classical Abstract Spaces in Functional Analysis 16
1.1 Introduction and Notation 16
1.2 Topological Spaces 20
1.3 Metric Spaces 36
1.4 Vector Spaces 56
1.5 Normed Vector Spaces 60
1.6 Space of Lebesgue Measurable Functions 67
1.7 Hilbert Spaces 73
Chapter 2 Linear Functionals and Linear Operators 78
2.1 Fundamental Theorems of Analysis 80
2.2 Dual Spaces 90
2.3 The Weak Topology 94
2.4 The Weak* Topology 95
2.5 Signed Measures and Topology 103
2.6 Riesz’s Representation Theorem 106
2.7 Closed Operators on Hilbert Spaces 110
2.8 Adjoint Operators 112
2.9 Gelfand Triples 118
2.10 Bilinear Mappings 121
Chapter 3 Common Function Spaces in Applications 126
3.1 The L Spaces 126
3.2 Sobolev Spaces 128
3.3 Banach Space Valued Functions 141
Chapter 4 Differential Calculus in Normed Vector Spaces 152
4.1 Differentiability of Functionals 152
4.2 Classical Examples of Differentiable Operators 158
Chapter 5 Minimization of Functionals 176
5.1 The Weierstrass Theorem 176
5.2 Elementary Calculus 178
5.3 Minimization of Differentiable Functionals 180
5.4 Equality Constrained Smooth Functionals 181
5.5 Frechet Differentiable Implicit Functionals 186
Chapter 6 Convex Functionals 190
6.1 Characterization of Convexity 192
6.2 Gateaux Differentiable Convex Functionals 195
6.3 Convex Programming in Rn 198
6.4 Ordered Vector Spaces 203
6.5 Convex Programming in Ordered Vector Spaces 208
6.6 Gateaux Di.erentiable Functionals on Ordered Vector Spaces 214
Chapter 7 Lower Semicontinuous Functionals 220
7.1 Characterization of Lower Semicontinuity 220
7.2 Lower Semicontinuous Functionals and Convexity 223
7.3 The Generalized Weierstrass Theorem 227
References 236
Index 238

Preface ( P. 12)

Overview of Book

This book evolved over a period of years as the authors taught classes in variational calculus and applied functional analysis to graduate students in engineering and mathematics. The book has likewise been influenced by the authors’ research programs that have relied on the application of functional analytic principles to problems in variational calculus, mechanics and control theory.

One of the most difficult tasks in preparing to utilize functional, convex, and set-valued analysis in practical problems in engineering and physics is the intimidating number of de.nitions, lemmas, theorems and propositions that constitute the foundations of functional analysis. It cannot be overemphasized that functional analysis can be a powerful tool for analyzing practical problems in mechanics and physics.

However, many academicians and researchers spend their lifetime studying abstract mathematics. It is a demanding field that requires discipline and devotion. It is a trite analogy that mathematics can be viewed as a pyramid of knowledge, that builds layer upon layer as more mathematical structure is put in place. The difficulty lies in the fact that an engineer or scientist typically would like to start somewhere "above the base" of the pyramid. Engineers and scientists are not as concerned, generally speaking, with the subtleties of deriving theorems axiomatically. Rather, they are interested in gaining a working knowledge of the applicability of the theory to their field of interest.

The content and structure of the book reffects the sometimes conflicting requirements of researchers or students who have formal training in either engineering or applied mathematics. Typically, before taking this course, those trained within an engineering discipline might have a working knowledge of fundamental topics in mechanics or control theory. Engineering students may be perfectly comfortable with the notion of the stress distribution in an elastic continuum, or the velocity field in an incompressible flow.

The formulation of the equations governing the static equilibrium of elastic bodies, or the structure of the Navier-Stokes Equations for incompressible flow, are often familiar to them. This is usually not the case for first year graduate students trained in applied mathematics. Rather, these students will have some familiarity with real analysis or functional analysis. The fundamental theorems of analysis including the Open Mapping Theorem, the Hahn-Banach Theorem, and the Closed Graph Theorem will constitute the foundations of their training in many cases.

Coupled with this essential disparity in the training to which graduate students in these two disciplines are exposed, it is a fact that formulations and solutions of modern problems in control and mechanics are couched in functional analytic terms. This trend is pervasive. Thus, the goal of the present text is admittedly ambitious. This text seeks to synthesize topics from abstract analysis with enough recent problems in control theory and mechanics to provide students from both disciplines with a working knowledge of functional analysis.

Organization

This work consists of two volumes. The primary thrust of this series is a discussion of how convex analysis, as a specific subtopic in functional analysis, has served to unify approaches in numerous problems in mechanics and control theory. Every attempt has been made to make the series self-contained. The first book in this series is dedicated to the fundamentals of convex functional analysis.