Kurdila / Zabarankin Convex Functional Analysis

1. Auflage 2006
ISBN: 978-3-7643-7357-3
Verlag: Springer
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 243 Seiten

Reihe: Systems & Control: Foundations & Applications

This volume is dedicated to the fundamentals of convex functional analysis. It presents those aspects of functional analysis that are extensively used in various applications to mechanics and control theory. The purpose of the text is essentially two-fold. On the one hand, a bare minimum of the theory required to understand the principles of functional, convex and set-valued analysis is presented.

Numerous examples and diagrams provide as intuitive an explanation of the principles as possible. On the other hand, the volume is largely self-contained. Those with a background in graduate mathematics will find a concise summary of all main definitions and theorems.

Written for:

Graduate students and researchers in functional analysis, approximation theory, convex analysis and control theory as well as engineers

Keywords:

Calculus of variations

Convex analysis

Functional analysis

Autoren/Hrsg.

Kurdila, Andrew J.

Zabarankin, Michael

Inhaltsverzeichnis

1;Contents;6
2;List of Figures;10
3;Preface;12
3.1;Overview of Book;12
3.2;Organization;13
3.3;Acknowledgements;14
4;Chapter 1 Classical Abstract Spaces in Functional Analysis;16
4.1;1.1 Introduction and Notation;16
4.2;1.2 Topological Spaces;20
4.3;1.3 Metric Spaces;36
4.4;1.4 Vector Spaces;56
4.5;1.5 Normed Vector Spaces;60
4.6;1.6 Space of Lebesgue Measurable Functions;67
4.7;1.7 Hilbert Spaces;73
5;Chapter 2 Linear Functionals and Linear Operators;78
5.1;2.1 Fundamental Theorems of Analysis;80
5.2;2.2 Dual Spaces;90
5.3;2.3 The Weak Topology;94
5.4;2.4 The Weak* Topology;95
5.5;2.5 Signed Measures and Topology;103
5.6;2.6 Riesz’s Representation Theorem;106
5.7;2.7 Closed Operators on Hilbert Spaces;110
5.8;2.8 Adjoint Operators;112
5.9;2.9 Gelfand Triples;118
5.10;2.10 Bilinear Mappings;121
6;Chapter 3 Common Function Spaces in Applications;126
6.1;3.1 The L Spaces;126
6.2;3.2 Sobolev Spaces;128
6.3;3.3 Banach Space Valued Functions;141
7;Chapter 4 Differential Calculus in Normed Vector Spaces;152
7.1;4.1 Differentiability of Functionals;152
7.2;4.2 Classical Examples of Differentiable Operators;158
8;Chapter 5 Minimization of Functionals;176
8.1;5.1 The Weierstrass Theorem;176
8.2;5.2 Elementary Calculus;178
8.3;5.3 Minimization of Differentiable Functionals;180
8.4;5.4 Equality Constrained Smooth Functionals;181
8.5;5.5 Frechet Differentiable Implicit Functionals;186
9;Chapter 6 Convex Functionals;190
9.1;6.1 Characterization of Convexity;192
9.2;6.2 Gateaux Differentiable Convex Functionals;195
9.3;6.3 Convex Programming in Rn;198
9.4;6.4 Ordered Vector Spaces;203
9.5;6.5 Convex Programming in Ordered Vector Spaces;208
9.6;6.6 Gateaux Di.erentiable Functionals on Ordered Vector Spaces;214
10;Chapter 7 Lower Semicontinuous Functionals;220
10.1;7.1 Characterization of Lower Semicontinuity;220
10.2;7.2 Lower Semicontinuous Functionals and Convexity;223
10.3;7.3 The Generalized Weierstrass Theorem;227
11;References;236
12;Index;238

Leseproben

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.